$sl{}\left(2\right)$-subalgebras of sp(16), type $C_8$

Number of sl(2) subalgebras: 99.

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Length longest root ambient algebra squared/4= 1/2

Given a root subsystem $P$, and a root sub-subsystem ${P}_{0}$, in (10.2) of Semisimple subalgebras of semisimple Lie algebras, E. Dynkin defines a numerical constant $e{}\left(P, {P}_{0}\right)$ (which we call Dynkin epsilon).
In Theorem 10.3, Dynkin proves that if an $sl{}\left(2\right)$ is an $S$-subalgebra in the root subalgebra generated by $P$, such that it has characteristic 2 for all simple roots of $P$ lying in ${P}_{0}$, then $e{}\left(P, {P}_{0}\right)=0$.

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H that wasn't realized on the first attempt but was ultimately realized: (26, 48, 66, 80, 90, 96, 98, 50), Type: ${A^{456}}_{1}$.
It turns out that in the current case of Cartan element h = (26, 48, 66, 80, 90, 96, 98, 50) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (22, 40, 54, 64, 70, 76, 78, 40), Type: ${A^{296}}_{1}$.
It turns out that in the current case of Cartan element h = (22, 40, 54, 64, 70, 76, 78, 40) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (22, 40, 54, 64, 70, 72, 74, 38), Type: ${A^{288}}_{1}$.
It turns out that in the current case of Cartan element h = (22, 40, 54, 64, 70, 72, 74, 38) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (22, 40, 54, 64, 70, 72, 74, 37), Type: ${A^{287}}_{1}$.
It turns out that in the current case of Cartan element h = (22, 40, 54, 64, 70, 72, 74, 37) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (18, 32, 42, 52, 58, 64, 66, 34), Type: ${A^{200}}_{1}$.
It turns out that in the current case of Cartan element h = (18, 32, 42, 52, 58, 64, 66, 34) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (18, 32, 42, 48, 54, 56, 58, 30), Type: ${A^{176}}_{1}$.
It turns out that in the current case of Cartan element h = (18, 32, 42, 48, 54, 56, 58, 30) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (18, 32, 42, 48, 54, 56, 58, 29), Type: ${A^{175}}_{1}$.
It turns out that in the current case of Cartan element h = (18, 32, 42, 48, 54, 56, 58, 29) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (18, 32, 42, 48, 50, 52, 54, 28), Type: ${A^{168}}_{1}$.
It turns out that in the current case of Cartan element h = (18, 32, 42, 48, 50, 52, 54, 28) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (18, 32, 42, 48, 50, 52, 54, 27), Type: ${A^{167}}_{1}$.
It turns out that in the current case of Cartan element h = (18, 32, 42, 48, 50, 52, 54, 27) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (18, 32, 42, 48, 50, 52, 52, 26), Type: ${A^{166}}_{1}$.
It turns out that in the current case of Cartan element h = (18, 32, 42, 48, 50, 52, 52, 26) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 34, 40, 46, 48, 50, 26), Type: ${A^{120}}_{1}$.
It turns out that in the current case of Cartan element h = (14, 24, 34, 40, 46, 48, 50, 26) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 34, 40, 46, 48, 50, 25), Type: ${A^{119}}_{1}$.
It turns out that in the current case of Cartan element h = (14, 24, 34, 40, 46, 48, 50, 25) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 36, 42, 44, 46, 24), Type: ${A^{104}}_{1}$.
It turns out that in the current case of Cartan element h = (14, 24, 30, 36, 42, 44, 46, 24) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 36, 38, 40, 42, 22), Type: ${A^{96}}_{1}$.
It turns out that in the current case of Cartan element h = (14, 24, 30, 36, 38, 40, 42, 22) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 36, 38, 40, 42, 21), Type: ${A^{95}}_{1}$.
It turns out that in the current case of Cartan element h = (14, 24, 30, 36, 38, 40, 42, 21) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 36, 38, 40, 40, 20), Type: ${A^{94}}_{1}$.
It turns out that in the current case of Cartan element h = (14, 24, 30, 36, 38, 40, 40, 20) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 34, 38, 40, 42, 21), Type: ${A^{93}}_{1}$.
It turns out that in the current case of Cartan element h = (14, 24, 30, 34, 38, 40, 42, 21) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 32, 34, 36, 38, 20), Type: ${A^{88}}_{1}$.
It turns out that in the current case of Cartan element h = (14, 24, 30, 32, 34, 36, 38, 20) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 32, 34, 36, 38, 19), Type: ${A^{87}}_{1}$.
It turns out that in the current case of Cartan element h = (14, 24, 30, 32, 34, 36, 38, 19) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 32, 34, 36, 36, 18), Type: ${A^{86}}_{1}$.
It turns out that in the current case of Cartan element h = (14, 24, 30, 32, 34, 36, 36, 18) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 32, 34, 34, 34, 17), Type: ${A^{85}}_{1}$.
It turns out that in the current case of Cartan element h = (14, 24, 30, 32, 34, 34, 34, 17) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 20, 26, 32, 38, 40, 42, 22), Type: ${A^{80}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 20, 26, 32, 38, 40, 42, 22) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 20, 26, 32, 34, 36, 38, 20), Type: ${A^{72}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 20, 26, 32, 34, 36, 38, 20) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 20, 26, 32, 34, 36, 38, 19), Type: ${A^{71}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 20, 26, 32, 34, 36, 38, 19) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 22, 28, 30, 32, 34, 18), Type: ${A^{56}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 16, 22, 28, 30, 32, 34, 18) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 22, 28, 30, 32, 34, 17), Type: ${A^{55}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 16, 22, 28, 30, 32, 34, 17) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 22, 26, 30, 32, 34, 17), Type: ${A^{53}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 16, 22, 26, 30, 32, 34, 17) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (8, 16, 22, 26, 30, 32, 34, 17), Type: ${A^{51}}_{1}$.
It turns out that in the current case of Cartan element h = (8, 16, 22, 26, 30, 32, 34, 17) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 22, 24, 26, 28, 30, 16), Type: ${A^{48}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 16, 22, 24, 26, 28, 30, 16) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 22, 24, 26, 28, 30, 15), Type: ${A^{47}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 16, 22, 24, 26, 28, 30, 15) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 22, 24, 26, 28, 28, 14), Type: ${A^{46}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 16, 22, 24, 26, 28, 28, 14) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 22, 24, 26, 26, 26, 13), Type: ${A^{45}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 16, 22, 24, 26, 26, 26, 13) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 20, 24, 26, 28, 30, 15), Type: ${A^{45}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 16, 20, 24, 26, 28, 30, 15) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 20, 24, 26, 28, 28, 14), Type: ${A^{44}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 16, 20, 24, 26, 28, 28, 14) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 18, 20, 22, 24, 26, 13), Type: ${A^{39}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 16, 18, 20, 22, 24, 26, 13) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 18, 20, 22, 24, 24, 12), Type: ${A^{38}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 16, 18, 20, 22, 24, 24, 12) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 18, 20, 22, 22, 22, 11), Type: ${A^{37}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 16, 18, 20, 22, 22, 22, 11) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 18, 20, 20, 20, 20, 10), Type: ${A^{36}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 16, 18, 20, 20, 20, 20, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 12, 18, 20, 22, 24, 26, 14), Type: ${A^{32}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 12, 18, 20, 22, 24, 26, 14) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 12, 18, 20, 22, 24, 26, 13), Type: ${A^{31}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 12, 18, 20, 22, 24, 26, 13) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 12, 16, 20, 22, 24, 26, 13), Type: ${A^{29}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 12, 16, 20, 22, 24, 26, 13) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 12, 14, 16, 18, 20, 22, 12), Type: ${A^{24}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 12, 14, 16, 18, 20, 22, 12) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 12, 14, 16, 18, 20, 22, 11), Type: ${A^{23}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 12, 14, 16, 18, 20, 22, 11) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 12, 14, 16, 18, 20, 20, 10), Type: ${A^{22}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 12, 14, 16, 18, 20, 20, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 12, 14, 16, 18, 18, 18, 9), Type: ${A^{21}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 12, 14, 16, 18, 18, 18, 9) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 10, 14, 16, 18, 20, 22, 11), Type: ${A^{21}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 10, 14, 16, 18, 20, 22, 11) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 10, 14, 16, 18, 20, 20, 10), Type: ${A^{20}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 10, 14, 16, 18, 20, 20, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 10, 14, 16, 18, 18, 18, 9), Type: ${A^{19}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 10, 14, 16, 18, 18, 18, 9) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 8, 10, 12, 14, 16, 18, 10), Type: ${A^{16}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 8, 10, 12, 14, 16, 18, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 8, 10, 12, 14, 16, 18, 9), Type: ${A^{15}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 8, 10, 12, 14, 16, 18, 9) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 8, 10, 12, 14, 16, 16, 8), Type: ${A^{14}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 8, 10, 12, 14, 16, 16, 8) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (4, 8, 10, 12, 14, 16, 18, 9), Type: ${A^{13}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 8, 10, 12, 14, 16, 18, 9) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 8, 10, 12, 14, 14, 14, 7), Type: ${A^{13}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 8, 10, 12, 14, 14, 14, 7) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 8, 10, 12, 12, 12, 12, 6), Type: ${A^{12}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 8, 10, 12, 12, 12, 12, 6) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (6, 8, 10, 10, 10, 10, 10, 5), Type: ${A^{11}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 8, 10, 10, 10, 10, 10, 5) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (4, 8, 10, 12, 14, 14, 14, 7), Type: ${A^{11}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 8, 10, 12, 14, 14, 14, 7) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

---

H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 8, 10, 12, 14, 8), Type: ${A^{8}}_{1}$.
It turns out that in the current case of Cartan element h = (2, 4, 6, 8, 10, 12, 14, 8) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 8, 10, 12, 14, 7), Type: ${A^{7}}_{1}$.
It turns out that in the current case of Cartan element h = (2, 4, 6, 8, 10, 12, 14, 7) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 8, 10, 12, 12, 6), Type: ${A^{6}}_{1}$.
It turns out that in the current case of Cartan element h = (2, 4, 6, 8, 10, 12, 12, 6) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 8, 10, 10, 10, 5), Type: ${A^{5}}_{1}$.
It turns out that in the current case of Cartan element h = (2, 4, 6, 8, 10, 10, 10, 5) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 8, 8, 8, 8, 4), Type: ${A^{4}}_{1}$.
It turns out that in the current case of Cartan element h = (2, 4, 6, 8, 8, 8, 8, 4) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 6, 6, 6, 6, 3), Type: ${A^{3}}_{1}$.
It turns out that in the current case of Cartan element h = (2, 4, 6, 6, 6, 6, 6, 3) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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Extensions of the rationals used (8 total): $\mathbb{Q}$$\mathbb{Q}$, $\mathbb{Q}[\sqrt{-1},\sqrt{3}]$$\mathbb{Q} [\sqrt{-1},\sqrt{3}]$, $\mathbb{Q}[\sqrt{-1},\sqrt{2}]$$\mathbb{Q} [\sqrt{-1},\sqrt{2}]$, $\(\mathbb Q[\sqrt{-1}, \sqrt{2}, \sqrt{3}]\)$, $\mathbb{Q}[\sqrt{-1},\sqrt{74}]$$\mathbb{Q} [\sqrt{-1},\sqrt{74}]$, $\(\mathbb Q[\sqrt{5}]\)$, $\(\mathbb Q[\sqrt{-1}]\)$, $\(\mathbb Q[\sqrt{-1}, \sqrt{2}, \sqrt{5}]\)$

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$A^{680}_1$
h-characteristic: (2, 2, 2, 2, 2, 2, 2, 2)
Length of the weight dual to h: 1360
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $C^{1}_8$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{30\omega_{1}}+V_{26\omega_{1}}+V_{22\omega_{1}}+V_{18\omega_{1}}+V_{14\omega_{1}}+V_{10\omega_{1}}+V_{6\omega_{1}}+V_{2\omega_{1}}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&64h_{8}+126h_{7}+120h_{6}+110h_{5}+96h_{4}+78h_{3}+56h_{2}+30h_{1}\\ e&=&64g_{8}+63g_{7}+60g_{6}+55g_{5}+48g_{4}+39g_{3}+28g_{2}+15g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-6}+g_{-7}+g_{-8}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&64h_{8}+126h_{7}+120h_{6}+110h_{5}+96h_{4}+78h_{3}+56h_{2}+30h_{1}\\ [h, e]&=&128g_{8}+126g_{7}+120g_{6}+110g_{5}+96g_{4}+78g_{3}+56g_{2}+30g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-5}-2g_{-6}-2g_{-7}-2g_{-8}\end{array}$
Centralizer type: $0$
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$A^{456}_1$
h-characteristic: (2, 2, 2, 2, 2, 2, 0, 2)
Length of the weight dual to h: 912
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_7+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_8$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{26\omega_{1}}+V_{22\omega_{1}}+V_{18\omega_{1}}+2V_{14\omega_{1}}+V_{12\omega_{1}}+V_{10\omega_{1}}+V_{6\omega_{1}}+2V_{2\omega_{1}}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&50h_{8}+98h_{7}+96h_{6}+90h_{5}+80h_{4}+66h_{3}+48h_{2}+26h_{1}\\ e&=&g_{22}+48g_{14}+49g_{8}+45g_{5}+40g_{4}+33g_{3}+24g_{2}+13g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-8}+g_{-14}+g_{-22}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&50h_{8}+98h_{7}+96h_{6}+90h_{5}+80h_{4}+66h_{3}+48h_{2}+26h_{1}\\ [h, e]&=&2g_{22}+96g_{14}+98g_{8}+90g_{5}+80g_{4}+66g_{3}+48g_{2}+26g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-5}-2g_{-8}-2g_{-14}-2g_{-22}\end{array}$
Centralizer type: $0$
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$A^{455}_1$
h-characteristic: (2, 2, 2, 2, 2, 2, 1, 0)
Length of the weight dual to h: 910
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $C^{1}_7$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{26\omega_{1}}+V_{22\omega_{1}}+V_{18\omega_{1}}+V_{14\omega_{1}}+2V_{13\omega_{1}}+V_{10\omega_{1}}+V_{6\omega_{1}}+V_{2\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&49h_{8}+98h_{7}+96h_{6}+90h_{5}+80h_{4}+66h_{3}+48h_{2}+26h_{1}\\ e&=&49g_{22}+48g_{6}+45g_{5}+40g_{4}+33g_{3}+24g_{2}+13g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-6}+g_{-22}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&49h_{8}+98h_{7}+96h_{6}+90h_{5}+80h_{4}+66h_{3}+48h_{2}+26h_{1}\\ [h, e]&=&98g_{22}+96g_{6}+90g_{5}+80g_{4}+66g_{3}+48g_{2}+26g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-5}-2g_{-6}-2g_{-22}\end{array}$
Centralizer type: $A_1$
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$A^{296}_1$
h-characteristic: (2, 2, 2, 2, 0, 2, 0, 2)
Length of the weight dual to h: 592
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_6+B^{1}_2$ Containing regular semisimple subalgebra number 2: $C^{1}_8$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{22\omega_{1}}+V_{18\omega_{1}}+2V_{14\omega_{1}}+V_{12\omega_{1}}+2V_{10\omega_{1}}+V_{8\omega_{1}}+2V_{6\omega_{1}}+2V_{2\omega_{1}}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&40h_{8}+78h_{7}+76h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ e&=&4g_{22}+35g_{14}+3g_{13}+32g_{12}+36g_{8}+27g_{3}+20g_{2}+11g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-8}+g_{-12}+g_{-13}+g_{-14}+g_{-22}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&40h_{8}+78h_{7}+76h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ [h, e]&=&8g_{22}+70g_{14}+6g_{13}+64g_{12}+72g_{8}+54g_{3}+40g_{2}+22g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-8}-2g_{-12}-2g_{-13}-2g_{-14}-2g_{-22}\end{array}$
Centralizer type: $0$
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$A^{288}_1$
h-characteristic: (2, 2, 2, 2, 2, 0, 0, 2)
Length of the weight dual to h: 576
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $C^{1}_6+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_6+A^{2}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_7+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{22\omega_{1}}+V_{18\omega_{1}}+V_{14\omega_{1}}+2V_{12\omega_{1}}+3V_{10\omega_{1}}+V_{6\omega_{1}}+4V_{2\omega_{1}}+V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&38h_{8}+74h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ e&=&g_{34}+g_{22}+35g_{20}+36g_{8}+32g_{4}+27g_{3}+20g_{2}+11g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-8}+g_{-20}+g_{-22}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&38h_{8}+74h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ [h, e]&=&2g_{34}+2g_{22}+70g_{20}+72g_{8}+64g_{4}+54g_{3}+40g_{2}+22g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-8}-2g_{-20}-2g_{-22}-2g_{-34}\end{array}$
Centralizer type: $0$
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$A^{287}_1$
h-characteristic: (2, 2, 2, 2, 2, 0, 1, 0)
Length of the weight dual to h: 574
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_6+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_7$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{22\omega_{1}}+V_{18\omega_{1}}+V_{14\omega_{1}}+V_{12\omega_{1}}+2V_{11\omega_{1}}+2V_{10\omega_{1}}+V_{6\omega_{1}}+2V_{2\omega_{1}}+2V_{\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&37h_{8}+74h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ e&=&g_{34}+36g_{22}+35g_{13}+32g_{4}+27g_{3}+20g_{2}+11g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-13}+g_{-22}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&37h_{8}+74h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ [h, e]&=&2g_{34}+72g_{22}+70g_{13}+64g_{4}+54g_{3}+40g_{2}+22g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-13}-2g_{-22}-2g_{-34}\end{array}$
Centralizer type: $A_1$
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$A^{286}_1$
h-characteristic: (2, 2, 2, 2, 2, 1, 0, 0)
Length of the weight dual to h: 572
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $C^{1}_6$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{22\omega_{1}}+V_{18\omega_{1}}+V_{14\omega_{1}}+4V_{11\omega_{1}}+V_{10\omega_{1}}+V_{6\omega_{1}}+V_{2\omega_{1}}+10V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&36h_{8}+72h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ e&=&36g_{34}+35g_{5}+32g_{4}+27g_{3}+20g_{2}+11g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&36h_{8}+72h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ [h, e]&=&72g_{34}+70g_{5}+64g_{4}+54g_{3}+40g_{2}+22g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-5}-2g_{-34}\end{array}$
Centralizer type: $B_2$
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$A^{200}_1$
h-characteristic: (2, 2, 0, 2, 0, 2, 0, 2)
Length of the weight dual to h: 400
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_5+C^{1}_3$ Containing regular semisimple subalgebra number 2: $C^{1}_8$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{18\omega_{1}}+2V_{14\omega_{1}}+V_{12\omega_{1}}+3V_{10\omega_{1}}+V_{8\omega_{1}}+3V_{6\omega_{1}}+V_{4\omega_{1}}+2V_{2\omega_{1}}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&34h_{8}+66h_{7}+64h_{6}+58h_{5}+52h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&9g_{22}+24g_{14}+8g_{13}+21g_{12}+5g_{11}+16g_{10}+25g_{8}+9g_{1}\\ f&=&g_{-1}+g_{-8}+g_{-10}+g_{-11}+g_{-12}+g_{-13}+g_{-14}+g_{-22}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&34h_{8}+66h_{7}+64h_{6}+58h_{5}+52h_{4}+42h_{3}+32h_{2}+18h_{1}\\ [h, e]&=&18g_{22}+48g_{14}+16g_{13}+42g_{12}+10g_{11}+32g_{10}+50g_{8}+18g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-8}-2g_{-10}-2g_{-11}-2g_{-12}-2g_{-13}-2g_{-14}-2g_{-22}\end{array}$
Centralizer type: $0$
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$A^{176}_1$
h-characteristic: (2, 2, 2, 0, 2, 0, 0, 2)
Length of the weight dual to h: 352
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $C^{1}_5+B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_8$ Containing regular semisimple subalgebra number 3: $C^{1}_7+A^{1}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_6+B^{1}_2$ Containing regular semisimple subalgebra number 5: $C^{1}_5+C^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{18\omega_{1}}+V_{14\omega_{1}}+V_{12\omega_{1}}+3V_{10\omega_{1}}+2V_{8\omega_{1}}+3V_{6\omega_{1}}+V_{4\omega_{1}}+4V_{2\omega_{1}}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&30h_{8}+58h_{7}+56h_{6}+54h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&4g_{34}+g_{22}+24g_{20}+3g_{12}+21g_{11}+25g_{8}+16g_{2}+9g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-8}+g_{-11}+g_{-12}+g_{-20}+g_{-22}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&30h_{8}+58h_{7}+56h_{6}+54h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ [h, e]&=&8g_{34}+2g_{22}+48g_{20}+6g_{12}+42g_{11}+50g_{8}+32g_{2}+18g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-8}-2g_{-11}-2g_{-12}-2g_{-20}-2g_{-22}-2g_{-34}\end{array}$
Centralizer type: $0$
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$A^{175}_1$
h-characteristic: (2, 2, 2, 0, 2, 0, 1, 0)
Length of the weight dual to h: 350
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_5+B^{1}_2$ Containing regular semisimple subalgebra number 2: $C^{1}_7$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{18\omega_{1}}+V_{14\omega_{1}}+V_{12\omega_{1}}+2V_{10\omega_{1}}+2V_{9\omega_{1}}+V_{8\omega_{1}}+3V_{6\omega_{1}}+2V_{3\omega_{1}}+2V_{2\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&29h_{8}+58h_{7}+56h_{6}+54h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&4g_{34}+25g_{22}+24g_{13}+3g_{12}+21g_{11}+16g_{2}+9g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-11}+g_{-12}+g_{-13}+g_{-22}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&29h_{8}+58h_{7}+56h_{6}+54h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ [h, e]&=&8g_{34}+50g_{22}+48g_{13}+6g_{12}+42g_{11}+32g_{2}+18g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-11}-2g_{-12}-2g_{-13}-2g_{-22}-2g_{-34}\end{array}$
Centralizer type: $A_1$
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$A^{173}_1$
h-characteristic: (2, 2, 2, 1, 0, 1, 1, 0)
Length of the weight dual to h: 346
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $C^{1}_5+A^{2}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{18\omega_{1}}+V_{14\omega_{1}}+2V_{11\omega_{1}}+V_{10\omega_{1}}+2V_{9\omega_{1}}+2V_{7\omega_{1}}+V_{6\omega_{1}}+3V_{4\omega_{1}}+2V_{2\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&29h_{8}+58h_{7}+56h_{6}+52h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&2g_{27}+25g_{22}+24g_{19}+2g_{14}+21g_{3}+16g_{2}+9g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-14}+g_{-19}+g_{-22}+g_{-27}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&29h_{8}+58h_{7}+56h_{6}+52h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ [h, e]&=&4g_{27}+50g_{22}+48g_{19}+4g_{14}+42g_{3}+32g_{2}+18g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-14}-2g_{-19}-2g_{-22}-2g_{-27}\end{array}$
Centralizer type: $A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 3): $h_{8}-2h_{5}$, $g_{5}+g_{-8}$, $g_{8}+g_{-5}$
Basis of centralizer intersected with cartan (dimension: 1): $1/2h_{8}-h_{5}$
Cartan of centralizer (dimension: 1): $1/2h_{8}-h_{5}$
Cartan-generating semisimple element: $1/2h_{8}-h_{5}$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^3-x$
Factorization of characteristic polynomial of ad H: (x )(x -1)(x +1)
Eigenvalues of ad H: $0$, $1$, $-1$
3 eigenvectors of ad H: $(1,0,0)$, $(0,0,1)$, $(0,1,0)$
Centralizer type: A^{3}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}-2h_{5}$
matching e: $g_{8}+g_{-5}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}-2h_{5}$
matching e: $g_{8}+g_{-5}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
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$A^{168}_1$
h-characteristic: (2, 2, 2, 2, 0, 0, 0, 2)
Length of the weight dual to h: 336
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $C^{1}_5+3A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_5+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_6+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_6+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{18\omega_{1}}+V_{14\omega_{1}}+4V_{10\omega_{1}}+3V_{8\omega_{1}}+V_{6\omega_{1}}+7V_{2\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&28h_{8}+54h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&g_{44}+g_{34}+24g_{26}+g_{22}+25g_{8}+21g_{3}+16g_{2}+9g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-8}+g_{-22}+g_{-26}+g_{-34}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&28h_{8}+54h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ [h, e]&=&2g_{44}+2g_{34}+48g_{26}+2g_{22}+50g_{8}+42g_{3}+32g_{2}+18g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-8}-2g_{-22}-2g_{-26}-2g_{-34}-2g_{-44}\end{array}$
Centralizer type: $A^{8}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 3): $g_{5}-g_{-5}$, $g_{6}-g_{-6}$, $g_{13}-g_{-13}$
Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 1): $g_{13}+g_{6}+g_{5}-g_{-5}-g_{-6}-g_{-13}$
Cartan-generating semisimple element: $g_{13}+g_{6}+g_{5}-g_{-5}-g_{-6}-g_{-13}$
adjoint action: $\begin{pmatrix}0 & -1 & 1\\ 1 & 0 & -1\\ -1 & 1 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^3+3x$
Factorization of characteristic polynomial of ad H: (x )(x^2+3)
Eigenvalues of ad H: $0$, $\sqrt{-3}$, $-\sqrt{-3}$
3 eigenvectors of ad H: $(1,1,1)$, $(−12−3−12,12−3−12,1)$, $(12−3−12,−12−3−12,1)$
Centralizer type: A^{8}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2/3\sqrt{-3}g_{13}-2/3\sqrt{-3}g_{6}-2/3\sqrt{-3}g_{5}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-6}+2/3\sqrt{-3}g_{-13}$
matching e: $g_{13}+\left(1/2\sqrt{-3}-1/2\right)g_{6}+\left(-1/2\sqrt{-3}-1/2\right)g_{5}+\left(1/2\sqrt{-3}+1/2\right)g_{-5}+\left(-1/2\sqrt{-3}+1/2\right)g_{-6}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 2/3\sqrt{-3} & -2/3\sqrt{-3}\\ -2/3\sqrt{-3} & 0 & 2/3\sqrt{-3}\\ 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2/3\sqrt{-3}g_{13}-2/3\sqrt{-3}g_{6}-2/3\sqrt{-3}g_{5}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-6}+2/3\sqrt{-3}g_{-13}$
matching e: $g_{13}+\left(1/2\sqrt{-3}-1/2\right)g_{6}+\left(-1/2\sqrt{-3}-1/2\right)g_{5}+\left(1/2\sqrt{-3}+1/2\right)g_{-5}+\left(-1/2\sqrt{-3}+1/2\right)g_{-6}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 2/3\sqrt{-3} & -2/3\sqrt{-3}\\ -2/3\sqrt{-3} & 0 & 2/3\sqrt{-3}\\ 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}288\\ \end{pmatrix}$
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$A^{168}_1$
h-characteristic: (0, 2, 0, 2, 0, 2, 0, 2)
Length of the weight dual to h: 336
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $2C^{1}_4$ Containing regular semisimple subalgebra number 2: $A^{2}_7$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{14\omega_{1}}+V_{12\omega_{1}}+3V_{10\omega_{1}}+V_{8\omega_{1}}+3V_{6\omega_{1}}+V_{4\omega_{1}}+3V_{2\omega_{1}}+V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&32h_{8}+62h_{7}+60h_{6}+54h_{5}+48h_{4}+38h_{3}+28h_{2}+14h_{1}\\ e&=&16g_{22}+7g_{16}+15g_{14}+15g_{13}+12g_{12}+12g_{11}+16g_{8}+7g_{2}\\ f&=&g_{-2}+g_{-8}+g_{-11}+g_{-12}+g_{-13}+g_{-14}+g_{-16}+g_{-22}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&32h_{8}+62h_{7}+60h_{6}+54h_{5}+48h_{4}+38h_{3}+28h_{2}+14h_{1}\\ [h, e]&=&32g_{22}+14g_{16}+30g_{14}+30g_{13}+24g_{12}+24g_{11}+32g_{8}+14g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-8}-2g_{-11}-2g_{-12}-2g_{-13}-2g_{-14}-2g_{-16}-2g_{-22}\end{array}$
Centralizer type: $0$
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$A^{167}_1$
h-characteristic: (2, 2, 2, 2, 0, 0, 1, 0)
Length of the weight dual to h: 334
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $C^{1}_5+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_5+A^{2}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_6+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{18\omega_{1}}+V_{14\omega_{1}}+3V_{10\omega_{1}}+2V_{9\omega_{1}}+2V_{8\omega_{1}}+V_{6\omega_{1}}+4V_{2\omega_{1}}+4V_{\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&27h_{8}+54h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&g_{44}+g_{34}+25g_{22}+24g_{19}+21g_{3}+16g_{2}+9g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-19}+g_{-22}+g_{-34}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&27h_{8}+54h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ [h, e]&=&2g_{44}+2g_{34}+50g_{22}+48g_{19}+42g_{3}+32g_{2}+18g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-19}-2g_{-22}-2g_{-34}-2g_{-44}\end{array}$
Centralizer type: $A_1$
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$A^{166}_1$
h-characteristic: (2, 2, 2, 2, 0, 1, 0, 0)
Length of the weight dual to h: 332
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_5+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_6$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{18\omega_{1}}+V_{14\omega_{1}}+2V_{10\omega_{1}}+4V_{9\omega_{1}}+V_{8\omega_{1}}+V_{6\omega_{1}}+2V_{2\omega_{1}}+4V_{\omega_{1}}+10V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&26h_{8}+52h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&g_{44}+25g_{34}+24g_{12}+21g_{3}+16g_{2}+9g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-12}+g_{-34}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&26h_{8}+52h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ [h, e]&=&2g_{44}+50g_{34}+48g_{12}+42g_{3}+32g_{2}+18g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-12}-2g_{-34}-2g_{-44}\end{array}$
Centralizer type: $B_2$
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$A^{165}_1$
h-characteristic: (2, 2, 2, 2, 1, 0, 0, 0)
Length of the weight dual to h: 330
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $C^{1}_5$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{18\omega_{1}}+V_{14\omega_{1}}+V_{10\omega_{1}}+6V_{9\omega_{1}}+V_{6\omega_{1}}+V_{2\omega_{1}}+21V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&25h_{8}+50h_{7}+50h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&25g_{44}+24g_{4}+21g_{3}+16g_{2}+9g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&25h_{8}+50h_{7}+50h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ [h, e]&=&50g_{44}+48g_{4}+42g_{3}+32g_{2}+18g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-44}\end{array}$
Centralizer type: $C_3$
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$A^{120}_1$
h-characteristic: (2, 0, 2, 0, 2, 0, 0, 2)
Length of the weight dual to h: 240
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $C^{1}_4+C^{1}_3+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_8$ Containing regular semisimple subalgebra number 3: $C^{1}_7+A^{1}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_5+C^{1}_3$ Containing regular semisimple subalgebra number 5: $2C^{1}_4$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{12\omega_{1}}+3V_{10\omega_{1}}+2V_{8\omega_{1}}+5V_{6\omega_{1}}+2V_{4\omega_{1}}+4V_{2\omega_{1}}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&26h_{8}+50h_{7}+48h_{6}+46h_{5}+40h_{4}+34h_{3}+24h_{2}+14h_{1}\\ e&=&g_{34}+9g_{22}+15g_{20}+8g_{19}+12g_{11}+5g_{10}+7g_{9}+16g_{8}\\ f&=&g_{-8}+g_{-9}+g_{-10}+g_{-11}+g_{-19}+g_{-20}+g_{-22}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&26h_{8}+50h_{7}+48h_{6}+46h_{5}+40h_{4}+34h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&2g_{34}+18g_{22}+30g_{20}+16g_{19}+24g_{11}+10g_{10}+14g_{9}+32g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-9}-2g_{-10}-2g_{-11}-2g_{-19}-2g_{-20}-2g_{-22}-2g_{-34}\end{array}$
Centralizer type: $0$
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$A^{119}_1$
h-characteristic: (2, 0, 2, 0, 2, 0, 1, 0)
Length of the weight dual to h: 238
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_4+C^{1}_3$ Containing regular semisimple subalgebra number 2: $C^{1}_7$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{12\omega_{1}}+3V_{10\omega_{1}}+V_{8\omega_{1}}+2V_{7\omega_{1}}+3V_{6\omega_{1}}+2V_{5\omega_{1}}+V_{4\omega_{1}}+3V_{2\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&25h_{8}+50h_{7}+48h_{6}+46h_{5}+40h_{4}+34h_{3}+24h_{2}+14h_{1}\\ e&=&9g_{34}+16g_{22}+15g_{13}+8g_{12}+12g_{11}+5g_{10}+7g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-11}+g_{-12}+g_{-13}+g_{-22}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&25h_{8}+50h_{7}+48h_{6}+46h_{5}+40h_{4}+34h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&18g_{34}+32g_{22}+30g_{13}+16g_{12}+24g_{11}+10g_{10}+14g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-11}-2g_{-12}-2g_{-13}-2g_{-22}-2g_{-34}\end{array}$
Centralizer type: $A_1$
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$A^{113}_1$
h-characteristic: (0, 2, 0, 2, 0, 1, 1, 0)
Length of the weight dual to h: 226
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{2}_6+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{12\omega_{1}}+V_{10\omega_{1}}+3V_{8\omega_{1}}+2V_{7\omega_{1}}+V_{6\omega_{1}}+2V_{5\omega_{1}}+3V_{4\omega_{1}}+2V_{2\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&25h_{8}+50h_{7}+48h_{6}+44h_{5}+40h_{4}+32h_{3}+24h_{2}+12h_{1}\\ e&=&g_{22}+12g_{21}+12g_{20}+6g_{16}+10g_{12}+10g_{11}+6g_{2}\\ f&=&g_{-2}+g_{-11}+g_{-12}+g_{-16}+g_{-20}+g_{-21}+g_{-22}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&25h_{8}+50h_{7}+48h_{6}+44h_{5}+40h_{4}+32h_{3}+24h_{2}+12h_{1}\\ [h, e]&=&2g_{22}+24g_{21}+24g_{20}+12g_{16}+20g_{12}+20g_{11}+12g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-11}-2g_{-12}-2g_{-16}-2g_{-20}-2g_{-21}-2g_{-22}\end{array}$
Centralizer type: $A^{7}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 3): $h_{8}+2h_{5}+2h_{3}-2h_{1}$, $g_{1}+g_{-3}+g_{-5}+g_{-8}$, $g_{8}+g_{5}+g_{3}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 1): $1/2h_{8}+h_{5}+h_{3}-h_{1}$
Cartan of centralizer (dimension: 1): $1/2h_{8}+h_{5}+h_{3}-h_{1}$
Cartan-generating semisimple element: $1/2h_{8}+h_{5}+h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^3-x$
Factorization of characteristic polynomial of ad H: (x )(x -1)(x +1)
Eigenvalues of ad H: $0$, $1$, $-1$
3 eigenvectors of ad H: $(1,0,0)$, $(0,0,1)$, $(0,1,0)$
Centralizer type: A^{7}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/63\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{5}+2h_{3}-2h_{1}$
matching e: $g_{8}+g_{5}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{5}+2h_{3}-2h_{1}$
matching e: $g_{8}+g_{5}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}252\\ \end{pmatrix}$
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$A^{112}_1$
h-characteristic: (0, 2, 0, 2, 0, 2, 0, 0)
Length of the weight dual to h: 224
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{2}_6$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{12\omega_{1}}+V_{10\omega_{1}}+3V_{8\omega_{1}}+5V_{6\omega_{1}}+3V_{4\omega_{1}}+V_{2\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&24h_{8}+48h_{7}+48h_{6}+44h_{5}+40h_{4}+32h_{3}+24h_{2}+12h_{1}\\ e&=&12g_{28}+6g_{16}+12g_{13}+10g_{12}+10g_{11}+6g_{2}\\ f&=&g_{-2}+g_{-11}+g_{-12}+g_{-13}+g_{-16}+g_{-28}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&24h_{8}+48h_{7}+48h_{6}+44h_{5}+40h_{4}+32h_{3}+24h_{2}+12h_{1}\\ [h, e]&=&24g_{28}+12g_{16}+24g_{13}+20g_{12}+20g_{11}+12g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-11}-2g_{-12}-2g_{-13}-2g_{-16}-2g_{-28}\end{array}$
Centralizer type: $A^{7}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 6): $g_{-8}$, $h_{7}+h_{5}+h_{3}-h_{1}$, $h_{8}$, $g_{1}+g_{-3}+g_{-5}+g_{-22}$, $g_{8}$, $g_{22}+g_{5}+g_{3}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{7}+h_{5}+h_{3}-h_{1}$, $-h_{8}$
Cartan of centralizer (dimension: 2): $h_{7}+h_{5}+h_{3}-h_{1}$, $-h_{8}$
Cartan-generating semisimple element: $-h_{8}+h_{7}+h_{5}+h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6-10x^4+9x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -3)(x -1)(x +1)(x +3)
Eigenvalues of ad H: $0$, $3$, $1$, $-1$, $-3$
6 eigenvectors of ad H: $(0,1,0,0,0,0)$, $(0,0,1,0,0,0)$, $(1,0,0,0,0,0)$, $(0,0,0,0,0,1)$, $(0,0,0,1,0,0)$, $(0,0,0,0,1,0)$
Centralizer type: A^{7}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/63\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}+2h_{5}+2h_{3}-2h_{1}$
matching e: $g_{22}+g_{5}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}+2h_{5}+2h_{3}-2h_{1}$
matching e: $g_{22}+g_{5}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}252\\ \end{pmatrix}$
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$A^{104}_1$
h-characteristic: (2, 2, 0, 0, 2, 0, 0, 2)
Length of the weight dual to h: 208
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $C^{1}_4+2B^{1}_2$ Containing regular semisimple subalgebra number 2: $C^{1}_4+A^{2}_3$ Containing regular semisimple subalgebra number 3: $C^{1}_6+B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+3V_{10\omega_{1}}+2V_{8\omega_{1}}+6V_{6\omega_{1}}+3V_{4\omega_{1}}+4V_{2\omega_{1}}+V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&24h_{8}+46h_{7}+44h_{6}+42h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&4g_{34}+4g_{22}+15g_{20}+3g_{19}+3g_{18}+12g_{17}+16g_{8}+7g_{1}\\ f&=&g_{-1}+g_{-8}+g_{-17}+g_{-18}+g_{-19}+g_{-20}+g_{-22}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&24h_{8}+46h_{7}+44h_{6}+42h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&8g_{34}+8g_{22}+30g_{20}+6g_{19}+6g_{18}+24g_{17}+32g_{8}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-8}-2g_{-17}-2g_{-18}-2g_{-19}-2g_{-20}-2g_{-22}-2g_{-34}\end{array}$
Centralizer type: $0$
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$A^{96}_1$
h-characteristic: (2, 2, 0, 2, 0, 0, 0, 2)
Length of the weight dual to h: 192
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 8
Containing regular semisimple subalgebra number 1: $C^{1}_4+B^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_4+B^{1}_2+A^{2}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_7+A^{1}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_6+2A^{1}_1$ Containing regular semisimple subalgebra number 5: $C^{1}_5+C^{1}_3$ Containing regular semisimple subalgebra number 6: $C^{1}_5+B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 7: $C^{1}_4+C^{1}_3+A^{1}_1$ Containing regular semisimple subalgebra number 8: $C^{1}_6+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+2V_{10\omega_{1}}+3V_{8\omega_{1}}+5V_{6\omega_{1}}+3V_{4\omega_{1}}+7V_{2\omega_{1}}+V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&22h_{8}+42h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&4g_{44}+g_{34}+15g_{26}+g_{22}+3g_{11}+12g_{10}+16g_{8}+7g_{1}\\ f&=&g_{-1}+g_{-8}+g_{-10}+g_{-11}+g_{-22}+g_{-26}+g_{-34}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&22h_{8}+42h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&8g_{44}+2g_{34}+30g_{26}+2g_{22}+6g_{11}+24g_{10}+32g_{8}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-8}-2g_{-10}-2g_{-11}-2g_{-22}-2g_{-26}-2g_{-34}-2g_{-44}\end{array}$
Centralizer type: $0$
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$A^{95}_1$
h-characteristic: (2, 2, 0, 2, 0, 0, 1, 0)
Length of the weight dual to h: 190
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $C^{1}_4+B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_7$ Containing regular semisimple subalgebra number 3: $C^{1}_6+A^{1}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_5+B^{1}_2$ Containing regular semisimple subalgebra number 5: $C^{1}_4+C^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+2V_{10\omega_{1}}+2V_{8\omega_{1}}+2V_{7\omega_{1}}+4V_{6\omega_{1}}+2V_{4\omega_{1}}+2V_{3\omega_{1}}+4V_{2\omega_{1}}+2V_{\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&21h_{8}+42h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&4g_{44}+g_{34}+16g_{22}+15g_{19}+3g_{11}+12g_{10}+7g_{1}\\ f&=&g_{-1}+g_{-10}+g_{-11}+g_{-19}+g_{-22}+g_{-34}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&21h_{8}+42h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&8g_{44}+2g_{34}+32g_{22}+30g_{19}+6g_{11}+24g_{10}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-10}-2g_{-11}-2g_{-19}-2g_{-22}-2g_{-34}-2g_{-44}\end{array}$
Centralizer type: $A_1$
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$A^{94}_1$
h-characteristic: (2, 2, 0, 2, 0, 1, 0, 0)
Length of the weight dual to h: 188
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_4+B^{1}_2$ Containing regular semisimple subalgebra number 2: $C^{1}_6$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+2V_{10\omega_{1}}+V_{8\omega_{1}}+4V_{7\omega_{1}}+3V_{6\omega_{1}}+V_{4\omega_{1}}+4V_{3\omega_{1}}+2V_{2\omega_{1}}+10V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&20h_{8}+40h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&4g_{44}+16g_{34}+15g_{12}+3g_{11}+12g_{10}+7g_{1}\\ f&=&g_{-1}+g_{-10}+g_{-11}+g_{-12}+g_{-34}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&20h_{8}+40h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&8g_{44}+32g_{34}+30g_{12}+6g_{11}+24g_{10}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-10}-2g_{-11}-2g_{-12}-2g_{-34}-2g_{-44}\end{array}$
Centralizer type: $B_2$
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$A^{93}_1$
h-characteristic: (2, 2, 1, 0, 1, 0, 1, 0)
Length of the weight dual to h: 186
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_4+A^{2}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_5+A^{2}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{10\omega_{1}}+2V_{9\omega_{1}}+V_{8\omega_{1}}+2V_{7\omega_{1}}+2V_{6\omega_{1}}+2V_{5\omega_{1}}+3V_{4\omega_{1}}+2V_{3\omega_{1}}+3V_{2\omega_{1}}+2V_{\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&21h_{8}+42h_{7}+40h_{6}+38h_{5}+34h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&g_{34}+2g_{32}+15g_{25}+16g_{22}+2g_{20}+12g_{2}+7g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-20}+g_{-22}+g_{-25}+g_{-32}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&21h_{8}+42h_{7}+40h_{6}+38h_{5}+34h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&2g_{34}+4g_{32}+30g_{25}+32g_{22}+4g_{20}+24g_{2}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-20}-2g_{-22}-2g_{-25}-2g_{-32}-2g_{-34}\end{array}$
Centralizer type: $A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 3): $h_{8}-2h_{4}$, $g_{4}+g_{-8}$, $g_{8}+g_{-4}$
Basis of centralizer intersected with cartan (dimension: 1): $1/2h_{8}-h_{4}$
Cartan of centralizer (dimension: 1): $1/2h_{8}-h_{4}$
Cartan-generating semisimple element: $1/2h_{8}-h_{4}$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^3-x$
Factorization of characteristic polynomial of ad H: (x )(x -1)(x +1)
Eigenvalues of ad H: $0$, $1$, $-1$
3 eigenvectors of ad H: $(1,0,0)$, $(0,0,1)$, $(0,1,0)$
Centralizer type: A^{3}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}-2h_{4}$
matching e: $g_{8}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}-2h_{4}$
matching e: $g_{8}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{92}_1$
h-characteristic: (2, 2, 1, 0, 1, 1, 0, 0)
Length of the weight dual to h: 184
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $C^{1}_4+A^{2}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{10\omega_{1}}+2V_{9\omega_{1}}+4V_{7\omega_{1}}+V_{6\omega_{1}}+2V_{5\omega_{1}}+3V_{4\omega_{1}}+6V_{2\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&20h_{8}+40h_{7}+40h_{6}+38h_{5}+34h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&2g_{38}+16g_{34}+15g_{18}+2g_{13}+12g_{2}+7g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-13}+g_{-18}+g_{-34}+g_{-38}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&20h_{8}+40h_{7}+40h_{6}+38h_{5}+34h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&4g_{38}+32g_{34}+30g_{18}+4g_{13}+24g_{2}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-13}-2g_{-18}-2g_{-34}-2g_{-38}\end{array}$
Centralizer type: $A^{3}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 6): $g_{-8}$, $h_{7}-h_{4}$, $h_{8}$, $g_{4}+g_{-22}$, $g_{8}$, $g_{22}+g_{-4}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{7}-h_{4}$, $-h_{8}$
Cartan of centralizer (dimension: 2): $h_{7}-h_{4}$, $-h_{8}$
Cartan-generating semisimple element: $-h_{8}+h_{7}-h_{4}$
adjoint action: $\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6-10x^4+9x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -3)(x -1)(x +1)(x +3)
Eigenvalues of ad H: $0$, $3$, $1$, $-1$, $-3$
6 eigenvectors of ad H: $(0,1,0,0,0,0)$, $(0,0,1,0,0,0)$, $(1,0,0,0,0,0)$, $(0,0,0,0,0,1)$, $(0,0,0,1,0,0)$, $(0,0,0,0,1,0)$
Centralizer type: A^{3}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}-2h_{4}$
matching e: $g_{22}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}-2h_{4}$
matching e: $g_{22}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
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$A^{88}_1$
h-characteristic: (2, 2, 2, 0, 0, 0, 0, 2)
Length of the weight dual to h: 176
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $C^{1}_4+4A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_4+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_4+2A^{2}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_5+3A^{1}_1$ Containing regular semisimple subalgebra number 5: $C^{1}_5+A^{2}_1+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{10\omega_{1}}+4V_{8\omega_{1}}+5V_{6\omega_{1}}+11V_{2\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&20h_{8}+38h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&g_{52}+g_{44}+g_{34}+15g_{31}+g_{22}+16g_{8}+12g_{2}+7g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-8}+g_{-22}+g_{-31}+g_{-34}+g_{-44}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&20h_{8}+38h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&2g_{52}+2g_{44}+2g_{34}+30g_{31}+2g_{22}+32g_{8}+24g_{2}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-8}-2g_{-22}-2g_{-31}-2g_{-34}-2g_{-44}-2g_{-52}\end{array}$
Centralizer type: $2A^{4}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 6): $g_{4}-g_{-4}$, $g_{5}-g_{-5}$, $g_{6}-g_{-6}$, $g_{12}-g_{-12}$, $g_{13}-g_{-13}$, $g_{19}-g_{-19}$
Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 2): $g_{19}+g_{6}+g_{5}+g_{4}-g_{-4}-g_{-5}-g_{-6}-g_{-19}$, $g_{13}+g_{12}-g_{-12}-g_{-13}$
Cartan-generating semisimple element: $g_{19}+g_{13}+g_{12}+g_{6}+g_{5}+g_{4}-g_{-4}-g_{-5}-g_{-6}-g_{-12}-g_{-13}-g_{-19}$
adjoint action: $\begin{pmatrix}0 & -1 & 0 & 1 & -1 & 1\\ 1 & 0 & -1 & -1 & 1 & 0\\ 0 & 1 & 0 & 1 & -1 & -1\\ -1 & 1 & -1 & 0 & 0 & 1\\ 1 & -1 & 1 & 0 & 0 & -1\\ -1 & 0 & 1 & -1 & 1 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6+12x^4+32x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x^2+4)(x^2+8)
Eigenvalues of ad H: $0$, $2\sqrt{-1}$, $-2\sqrt{-1}$, $2\sqrt{-2}$, $-2\sqrt{-2}$
6 eigenvectors of ad H: $(0,0,0,1,1,0)$, $(1,1,1,0,0,1)$, $(−−1,−1,−1,0,0,1)$, $(−1,−1,−−1,0,0,1)$, $(−1,1,−1,−−2,−2,1)$, $(−1,1,−1,−2,−−2,1)$
Centralizer type: 2A^{4}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-\sqrt{-1}g_{13}-\sqrt{-1}g_{12}+\sqrt{-1}g_{-12}+\sqrt{-1}g_{-13}$
matching e: $g_{19}+\sqrt{-1}g_{6}-g_{5}-\sqrt{-1}g_{4}+\sqrt{-1}g_{-4}+g_{-5}-\sqrt{-1}g_{-6}-g_{-19}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & \sqrt{-1} & 0 & 0 & 0 & -\sqrt{-1}\\ -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0\\ 0 & -\sqrt{-1} & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-\sqrt{-1}g_{13}-\sqrt{-1}g_{12}+\sqrt{-1}g_{-12}+\sqrt{-1}g_{-13}$
matching e: $g_{19}+\sqrt{-1}g_{6}-g_{5}-\sqrt{-1}g_{4}+\sqrt{-1}g_{-4}+g_{-5}-\sqrt{-1}g_{-6}-g_{-19}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & \sqrt{-1} & 0 & 0 & 0 & -\sqrt{-1}\\ -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0\\ 0 & -\sqrt{-1} & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-1/2\sqrt{-2}g_{19}-1/2\sqrt{-2}g_{6}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{4}+1/2\sqrt{-2}g_{-4}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-6}+1/2\sqrt{-2}g_{-19}$
matching e: $g_{19}+\sqrt{-2}g_{13}-\sqrt{-2}g_{12}-g_{6}+g_{5}-g_{4}+g_{-4}-g_{-5}+g_{-6}+\sqrt{-2}g_{-12}-\sqrt{-2}g_{-13}-g_{-19}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & -1/2\sqrt{-2}\\ -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 1/2\sqrt{-2}\\ 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-1/2\sqrt{-2}g_{19}-1/2\sqrt{-2}g_{6}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{4}+1/2\sqrt{-2}g_{-4}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-6}+1/2\sqrt{-2}g_{-19}$
matching e: $g_{19}+\sqrt{-2}g_{13}-\sqrt{-2}g_{12}-g_{6}+g_{5}-g_{4}+g_{-4}-g_{-5}+g_{-6}+\sqrt{-2}g_{-12}-\sqrt{-2}g_{-13}-g_{-19}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & -1/2\sqrt{-2}\\ -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 1/2\sqrt{-2}\\ 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{87}_1$
h-characteristic: (2, 2, 2, 0, 0, 0, 1, 0)
Length of the weight dual to h: 174
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $C^{1}_4+3A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_4+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_5+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_5+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{10\omega_{1}}+3V_{8\omega_{1}}+2V_{7\omega_{1}}+4V_{6\omega_{1}}+7V_{2\omega_{1}}+6V_{\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&19h_{8}+38h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&g_{52}+g_{44}+g_{34}+15g_{25}+16g_{22}+12g_{2}+7g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-22}+g_{-25}+g_{-34}+g_{-44}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&19h_{8}+38h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&2g_{52}+2g_{44}+2g_{34}+30g_{25}+32g_{22}+24g_{2}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-22}-2g_{-25}-2g_{-34}-2g_{-44}-2g_{-52}\end{array}$
Centralizer type: $A^{8}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 6): $g_{-8}$, $h_{8}$, $g_{4}-g_{-4}$, $g_{5}-g_{-5}$, $g_{8}$, $g_{12}-g_{-12}$
Basis of centralizer intersected with cartan (dimension: 1): $-h_{8}$
Cartan of centralizer (dimension: 2): $-g_{8}-h_{8}-g_{-8}$, $g_{12}+g_{5}+g_{4}-g_{-4}-g_{-5}-g_{-12}$
Cartan-generating semisimple element: $g_{12}+g_{8}+g_{5}+g_{4}+h_{8}-g_{-4}-g_{-5}+g_{-8}-g_{-12}$
adjoint action: $\begin{pmatrix}-2 & 2 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & -1\\ 0 & -2 & 0 & 0 & 2 & 0\\ 0 & 0 & -1 & 1 & 0 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6-5x^4-24x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x^2-8)(x^2+3)
Eigenvalues of ad H: $0$, $2\sqrt{2}$, $-2\sqrt{2}$, $\sqrt{-3}$, $-\sqrt{-3}$
6 eigenvectors of ad H: $(1,1,0,0,1,0)$, $(0,0,1,1,0,1)$, $(22−3,−2+1,0,0,1,0)$, $(−22−3,2+1,0,0,1,0)$, $(0,0,−12−3−12,12−3−12,0,1)$, $(0,0,12−3−12,−12−3−12,0,1)$
Centralizer type: A^{8}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$1/2\sqrt{2}g_{8}+1/2\sqrt{2}h_{8}+1/2\sqrt{2}g_{-8}$
matching e: $g_{8}+\left(-\sqrt{2}+1\right)h_{8}+\left(2\sqrt{2}-3\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & -1/2\sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$1/2\sqrt{2}g_{8}+1/2\sqrt{2}h_{8}+1/2\sqrt{2}g_{-8}$
matching e: $g_{8}+\left(-\sqrt{2}+1\right)h_{8}+\left(2\sqrt{2}-3\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & -1/2\sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2/3\sqrt{-3}g_{12}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-12}$
matching e: $g_{12}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2/3\sqrt{-3}g_{12}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-12}$
matching e: $g_{12}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}288\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{86}_1$
h-characteristic: (2, 2, 2, 0, 0, 1, 0, 0)
Length of the weight dual to h: 172
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $C^{1}_4+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_4+A^{2}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_5+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{10\omega_{1}}+2V_{8\omega_{1}}+4V_{7\omega_{1}}+3V_{6\omega_{1}}+4V_{2\omega_{1}}+8V_{\omega_{1}}+11V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&18h_{8}+36h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&g_{52}+g_{44}+16g_{34}+15g_{18}+12g_{2}+7g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-18}+g_{-34}+g_{-44}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&18h_{8}+36h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&2g_{52}+2g_{44}+32g_{34}+30g_{18}+24g_{2}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-18}-2g_{-34}-2g_{-44}-2g_{-52}\end{array}$
Centralizer type: $B_2$
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$A^{85}_1$
h-characteristic: (2, 2, 2, 0, 1, 0, 0, 0)
Length of the weight dual to h: 170
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_4+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_5$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{10\omega_{1}}+V_{8\omega_{1}}+6V_{7\omega_{1}}+2V_{6\omega_{1}}+2V_{2\omega_{1}}+6V_{\omega_{1}}+21V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&17h_{8}+34h_{7}+34h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&g_{52}+16g_{44}+15g_{11}+12g_{2}+7g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-11}+g_{-44}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&17h_{8}+34h_{7}+34h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&2g_{52}+32g_{44}+30g_{11}+24g_{2}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-11}-2g_{-44}-2g_{-52}\end{array}$
Centralizer type: $C_3$
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$A^{84}_1$
h-characteristic: (2, 2, 2, 1, 0, 0, 0, 0)
Length of the weight dual to h: 168
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $C^{1}_4$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{10\omega_{1}}+8V_{7\omega_{1}}+V_{6\omega_{1}}+V_{2\omega_{1}}+36V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&16h_{8}+32h_{7}+32h_{6}+32h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&16g_{52}+15g_{3}+12g_{2}+7g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&16h_{8}+32h_{7}+32h_{6}+32h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&32g_{52}+30g_{3}+24g_{2}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-52}\end{array}$
Centralizer type: $C_4$
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$A^{80}_1$
h-characteristic: (0, 2, 0, 0, 2, 0, 0, 2)
Length of the weight dual to h: 160
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $2C^{1}_3+B^{1}_2$ Containing regular semisimple subalgebra number 2: $A^{2}_5+B^{1}_2$ Containing regular semisimple subalgebra number 3: $C^{1}_5+C^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{10\omega_{1}}+3V_{8\omega_{1}}+6V_{6\omega_{1}}+3V_{4\omega_{1}}+6V_{2\omega_{1}}+V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&22h_{8}+42h_{7}+40h_{6}+38h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ e&=&4g_{34}+5g_{23}+9g_{22}+8g_{20}+8g_{19}+3g_{18}+5g_{10}+9g_{8}\\ f&=&g_{-8}+g_{-10}+g_{-18}+g_{-19}+g_{-20}+g_{-22}+g_{-23}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&22h_{8}+42h_{7}+40h_{6}+38h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ [h, e]&=&8g_{34}+10g_{23}+18g_{22}+16g_{20}+16g_{19}+6g_{18}+10g_{10}+18g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-10}-2g_{-18}-2g_{-19}-2g_{-20}-2g_{-22}-2g_{-23}-2g_{-34}\end{array}$
Centralizer type: $0$
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$A^{75}_1$
h-characteristic: (1, 0, 1, 1, 0, 1, 1, 0)
Length of the weight dual to h: 150
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{2}_4+C^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+2V_{9\omega_{1}}+3V_{8\omega_{1}}+2V_{7\omega_{1}}+2V_{6\omega_{1}}+2V_{5\omega_{1}}+3V_{4\omega_{1}}+2V_{3\omega_{1}}+2V_{2\omega_{1}}+2V_{\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&21h_{8}+42h_{7}+40h_{6}+36h_{5}+32h_{4}+26h_{3}+18h_{2}+10h_{1}\\ e&=&4g_{24}+9g_{22}+6g_{21}+6g_{20}+8g_{19}+5g_{16}+4g_{11}\\ f&=&g_{-11}+g_{-16}+g_{-19}+g_{-20}+g_{-21}+g_{-22}+g_{-24}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&21h_{8}+42h_{7}+40h_{6}+36h_{5}+32h_{4}+26h_{3}+18h_{2}+10h_{1}\\ [h, e]&=&8g_{24}+18g_{22}+12g_{21}+12g_{20}+16g_{19}+10g_{16}+8g_{11}\\ [h, f]&=&-2g_{-11}-2g_{-16}-2g_{-19}-2g_{-20}-2g_{-21}-2g_{-22}-2g_{-24}\end{array}$
Centralizer type: $A^{5}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 3): $h_{8}+2h_{5}-2h_{2}$, $g_{2}+g_{-5}+g_{-8}$, $g_{8}+g_{5}+g_{-2}$
Basis of centralizer intersected with cartan (dimension: 1): $1/2h_{8}+h_{5}-h_{2}$
Cartan of centralizer (dimension: 1): $1/2h_{8}+h_{5}-h_{2}$
Cartan-generating semisimple element: $1/2h_{8}+h_{5}-h_{2}$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^3-x$
Factorization of characteristic polynomial of ad H: (x )(x -1)(x +1)
Eigenvalues of ad H: $0$, $1$, $-1$
3 eigenvectors of ad H: $(1,0,0)$, $(0,0,1)$, $(0,1,0)$
Centralizer type: A^{5}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{5}-2h_{2}$
matching e: $g_{8}+g_{5}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{5}-2h_{2}$
matching e: $g_{8}+g_{5}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{72}_1$
h-characteristic: (0, 2, 0, 2, 0, 0, 0, 2)
Length of the weight dual to h: 144
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 6
Containing regular semisimple subalgebra number 1: $2C^{1}_3+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_5+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $2C^{1}_3+A^{2}_1$ Containing regular semisimple subalgebra number 4: $A^{2}_5+A^{2}_1$ Containing regular semisimple subalgebra number 5: $2C^{1}_4$ Containing regular semisimple subalgebra number 6: $C^{1}_4+C^{1}_3+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{10\omega_{1}}+V_{8\omega_{1}}+7V_{6\omega_{1}}+5V_{4\omega_{1}}+6V_{2\omega_{1}}+2V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&20h_{8}+38h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ e&=&g_{44}+g_{34}+8g_{26}+8g_{25}+9g_{22}+5g_{16}+9g_{8}+5g_{2}\\ f&=&g_{-2}+g_{-8}+g_{-16}+g_{-22}+g_{-25}+g_{-26}+g_{-34}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&20h_{8}+38h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ [h, e]&=&2g_{44}+2g_{34}+16g_{26}+16g_{25}+18g_{22}+10g_{16}+18g_{8}+10g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-8}-2g_{-16}-2g_{-22}-2g_{-25}-2g_{-26}-2g_{-34}-2g_{-44}\end{array}$
Centralizer type: $0$
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$A^{71}_1$
h-characteristic: (0, 2, 0, 2, 0, 0, 1, 0)
Length of the weight dual to h: 142
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $2C^{1}_3+A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_5+A^{1}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_4+C^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{10\omega_{1}}+V_{8\omega_{1}}+5V_{6\omega_{1}}+4V_{5\omega_{1}}+3V_{4\omega_{1}}+4V_{2\omega_{1}}+2V_{\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&19h_{8}+38h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ e&=&g_{44}+9g_{34}+9g_{22}+8g_{19}+8g_{18}+5g_{16}+5g_{2}\\ f&=&g_{-2}+g_{-16}+g_{-18}+g_{-19}+g_{-22}+g_{-34}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&19h_{8}+38h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ [h, e]&=&2g_{44}+18g_{34}+18g_{22}+16g_{19}+16g_{18}+10g_{16}+10g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-16}-2g_{-18}-2g_{-19}-2g_{-22}-2g_{-34}-2g_{-44}\end{array}$
Centralizer type: $A_1$
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$A^{70}_1$
h-characteristic: (0, 2, 0, 2, 0, 1, 0, 0)
Length of the weight dual to h: 140
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $2C^{1}_3$ Containing regular semisimple subalgebra number 2: $A^{2}_5$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{10\omega_{1}}+V_{8\omega_{1}}+3V_{6\omega_{1}}+8V_{5\omega_{1}}+V_{4\omega_{1}}+3V_{2\omega_{1}}+11V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&18h_{8}+36h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ e&=&9g_{44}+9g_{34}+5g_{16}+8g_{12}+8g_{11}+5g_{2}\\ f&=&g_{-2}+g_{-11}+g_{-12}+g_{-16}+g_{-34}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&18h_{8}+36h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ [h, e]&=&18g_{44}+18g_{34}+10g_{16}+16g_{12}+16g_{11}+10g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-11}-2g_{-12}-2g_{-16}-2g_{-34}-2g_{-44}\end{array}$
Centralizer type: $B_2$
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$A^{56}_1$
h-characteristic: (2, 0, 0, 2, 0, 0, 0, 2)
Length of the weight dual to h: 112
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 8
Containing regular semisimple subalgebra number 1: $C^{1}_3+2B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_3+C^{1}_3+A^{1}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_6+B^{1}_2$ Containing regular semisimple subalgebra number 4: $C^{1}_5+C^{1}_3$ Containing regular semisimple subalgebra number 5: $C^{1}_5+B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 6: $C^{1}_4+2B^{1}_2$ Containing regular semisimple subalgebra number 7: $2C^{1}_3+B^{1}_2$ Containing regular semisimple subalgebra number 8: $C^{1}_4+A^{2}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+2V_{8\omega_{1}}+7V_{6\omega_{1}}+6V_{4\omega_{1}}+9V_{2\omega_{1}}+V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&18h_{8}+34h_{7}+32h_{6}+30h_{5}+28h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&4g_{44}+4g_{34}+8g_{26}+g_{22}+3g_{18}+3g_{17}+5g_{16}+9g_{8}\\ f&=&g_{-8}+g_{-16}+g_{-17}+g_{-18}+g_{-22}+g_{-26}+g_{-34}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&18h_{8}+34h_{7}+32h_{6}+30h_{5}+28h_{4}+22h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&8g_{44}+8g_{34}+16g_{26}+2g_{22}+6g_{18}+6g_{17}+10g_{16}+18g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-16}-2g_{-17}-2g_{-18}-2g_{-22}-2g_{-26}-2g_{-34}-2g_{-44}\end{array}$
Centralizer type: $0$
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$A^{55}_1$
h-characteristic: (2, 0, 0, 2, 0, 0, 1, 0)
Length of the weight dual to h: 110
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $C^{1}_3+2B^{1}_2$ Containing regular semisimple subalgebra number 2: $A^{2}_3+C^{1}_3$ Containing regular semisimple subalgebra number 3: $C^{1}_5+B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+2V_{8\omega_{1}}+6V_{6\omega_{1}}+2V_{5\omega_{1}}+3V_{4\omega_{1}}+4V_{3\omega_{1}}+6V_{2\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&17h_{8}+34h_{7}+32h_{6}+30h_{5}+28h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&4g_{44}+4g_{34}+9g_{22}+8g_{19}+3g_{18}+3g_{17}+5g_{16}\\ f&=&g_{-16}+g_{-17}+g_{-18}+g_{-19}+g_{-22}+g_{-34}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&17h_{8}+34h_{7}+32h_{6}+30h_{5}+28h_{4}+22h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&8g_{44}+8g_{34}+18g_{22}+16g_{19}+6g_{18}+6g_{17}+10g_{16}\\ [h, f]&=&-2g_{-16}-2g_{-17}-2g_{-18}-2g_{-19}-2g_{-22}-2g_{-34}-2g_{-44}\end{array}$
Centralizer type: $A_1$
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$A^{53}_1$
h-characteristic: (2, 0, 1, 0, 1, 0, 1, 0)
Length of the weight dual to h: 106
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_3+A^{2}_2+B^{1}_2$ Containing regular semisimple subalgebra number 2: $C^{1}_5+A^{2}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+V_{8\omega_{1}}+2V_{7\omega_{1}}+3V_{6\omega_{1}}+4V_{5\omega_{1}}+4V_{4\omega_{1}}+4V_{3\omega_{1}}+4V_{2\omega_{1}}+2V_{\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&17h_{8}+34h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&4g_{34}+2g_{32}+8g_{25}+3g_{24}+9g_{22}+2g_{20}+5g_{9}\\ f&=&g_{-9}+g_{-20}+g_{-22}+g_{-24}+g_{-25}+g_{-32}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&17h_{8}+34h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&8g_{34}+4g_{32}+16g_{25}+6g_{24}+18g_{22}+4g_{20}+10g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-20}-2g_{-22}-2g_{-24}-2g_{-25}-2g_{-32}-2g_{-34}\end{array}$
Centralizer type: $A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 3): $h_{8}-2h_{4}$, $g_{4}+g_{-8}$, $g_{8}+g_{-4}$
Basis of centralizer intersected with cartan (dimension: 1): $1/2h_{8}-h_{4}$
Cartan of centralizer (dimension: 1): $1/2h_{8}-h_{4}$
Cartan-generating semisimple element: $1/2h_{8}-h_{4}$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^3-x$
Factorization of characteristic polynomial of ad H: (x )(x -1)(x +1)
Eigenvalues of ad H: $0$, $1$, $-1$
3 eigenvectors of ad H: $(1,0,0)$, $(0,0,1)$, $(0,1,0)$
Centralizer type: A^{3}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}-2h_{4}$
matching e: $g_{8}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}-2h_{4}$
matching e: $g_{8}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
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$A^{51}_1$
h-characteristic: (0, 1, 1, 0, 1, 0, 1, 0)
Length of the weight dual to h: 102
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{2}_4+B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_4+C^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{8\omega_{1}}+2V_{7\omega_{1}}+2V_{6\omega_{1}}+4V_{5\omega_{1}}+4V_{4\omega_{1}}+4V_{3\omega_{1}}+4V_{2\omega_{1}}+2V_{\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&17h_{8}+34h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+8h_{1}\\ e&=&4g_{34}+6g_{27}+6g_{26}+4g_{23}+g_{22}+3g_{18}+4g_{10}\\ f&=&g_{-10}+g_{-18}+g_{-22}+g_{-23}+g_{-26}+g_{-27}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&17h_{8}+34h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+8h_{1}\\ [h, e]&=&8g_{34}+12g_{27}+12g_{26}+8g_{23}+2g_{22}+6g_{18}+8g_{10}\\ [h, f]&=&-2g_{-10}-2g_{-18}-2g_{-22}-2g_{-23}-2g_{-26}-2g_{-27}-2g_{-34}\end{array}$
Centralizer type: $A^{5}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 3): $h_{8}+2h_{4}-2h_{1}$, $g_{1}+g_{-4}+g_{-8}$, $g_{8}+g_{4}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 1): $1/2h_{8}+h_{4}-h_{1}$
Cartan of centralizer (dimension: 1): $1/2h_{8}+h_{4}-h_{1}$
Cartan-generating semisimple element: $1/2h_{8}+h_{4}-h_{1}$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^3-x$
Factorization of characteristic polynomial of ad H: (x )(x -1)(x +1)
Eigenvalues of ad H: $0$, $1$, $-1$
3 eigenvectors of ad H: $(1,0,0)$, $(0,0,1)$, $(0,1,0)$
Centralizer type: A^{5}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{4}-2h_{1}$
matching e: $g_{8}+g_{4}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{4}-2h_{1}$
matching e: $g_{8}+g_{4}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
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$A^{50}_1$
h-characteristic: (0, 1, 1, 0, 1, 1, 0, 0)
Length of the weight dual to h: 100
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{2}_4+B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{8\omega_{1}}+2V_{7\omega_{1}}+2V_{6\omega_{1}}+2V_{5\omega_{1}}+7V_{4\omega_{1}}+4V_{3\omega_{1}}+2V_{2\omega_{1}}+2V_{\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&16h_{8}+32h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+8h_{1}\\ e&=&4g_{34}+6g_{33}+4g_{23}+6g_{19}+3g_{18}+4g_{10}\\ f&=&g_{-10}+g_{-18}+g_{-19}+g_{-23}+g_{-33}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&16h_{8}+32h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+8h_{1}\\ [h, e]&=&8g_{34}+12g_{33}+8g_{23}+12g_{19}+6g_{18}+8g_{10}\\ [h, f]&=&-2g_{-10}-2g_{-18}-2g_{-19}-2g_{-23}-2g_{-33}-2g_{-34}\end{array}$
Centralizer type: $A^{5}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 6): $g_{-8}$, $h_{7}+h_{4}-h_{1}$, $h_{8}$, $g_{1}+g_{-4}+g_{-22}$, $g_{8}$, $g_{22}+g_{4}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{7}+h_{4}-h_{1}$, $-h_{8}$
Cartan of centralizer (dimension: 2): $h_{7}+h_{4}-h_{1}$, $-h_{8}$
Cartan-generating semisimple element: $-h_{8}+h_{7}+h_{4}-h_{1}$
adjoint action: $\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6-10x^4+9x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -3)(x -1)(x +1)(x +3)
Eigenvalues of ad H: $0$, $3$, $1$, $-1$, $-3$
6 eigenvectors of ad H: $(0,1,0,0,0,0)$, $(0,0,1,0,0,0)$, $(1,0,0,0,0,0)$, $(0,0,0,0,0,1)$, $(0,0,0,1,0,0)$, $(0,0,0,0,1,0)$
Centralizer type: A^{5}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}+2h_{4}-2h_{1}$
matching e: $g_{22}+g_{4}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}+2h_{4}-2h_{1}$
matching e: $g_{22}+g_{4}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
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$A^{48}_1$
h-characteristic: (2, 0, 2, 0, 0, 0, 0, 2)
Length of the weight dual to h: 96
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 11
Containing regular semisimple subalgebra number 1: $C^{1}_3+B^{1}_2+3A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_3+B^{1}_2+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_6+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_5+3A^{1}_1$ Containing regular semisimple subalgebra number 5: $C^{1}_4+C^{1}_3+A^{1}_1$ Containing regular semisimple subalgebra number 6: $C^{1}_4+B^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 7: $2C^{1}_3+2A^{1}_1$ Containing regular semisimple subalgebra number 8: $C^{1}_6+A^{2}_1$ Containing regular semisimple subalgebra number 9: $C^{1}_5+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 10: $C^{1}_4+B^{1}_2+A^{2}_1$ Containing regular semisimple subalgebra number 11: $2C^{1}_3+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+V_{8\omega_{1}}+6V_{6\omega_{1}}+7V_{4\omega_{1}}+12V_{2\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&16h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&4g_{52}+g_{44}+g_{34}+8g_{31}+g_{22}+3g_{10}+5g_{9}+9g_{8}\\ f&=&g_{-8}+g_{-9}+g_{-10}+g_{-22}+g_{-31}+g_{-34}+g_{-44}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&16h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&8g_{52}+2g_{44}+2g_{34}+16g_{31}+2g_{22}+6g_{10}+10g_{9}+18g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-9}-2g_{-10}-2g_{-22}-2g_{-31}-2g_{-34}-2g_{-44}-2g_{-52}\end{array}$
Centralizer type: $A^{8}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 3): $g_{5}-g_{-5}$, $g_{6}-g_{-6}$, $g_{13}-g_{-13}$
Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 1): $g_{13}+g_{6}+g_{5}-g_{-5}-g_{-6}-g_{-13}$
Cartan-generating semisimple element: $g_{13}+g_{6}+g_{5}-g_{-5}-g_{-6}-g_{-13}$
adjoint action: $\begin{pmatrix}0 & -1 & 1\\ 1 & 0 & -1\\ -1 & 1 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^3+3x$
Factorization of characteristic polynomial of ad H: (x )(x^2+3)
Eigenvalues of ad H: $0$, $\sqrt{-3}$, $-\sqrt{-3}$
3 eigenvectors of ad H: $(1,1,1)$, $(−12−3−12,12−3−12,1)$, $(12−3−12,−12−3−12,1)$
Centralizer type: A^{8}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2/3\sqrt{-3}g_{13}-2/3\sqrt{-3}g_{6}-2/3\sqrt{-3}g_{5}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-6}+2/3\sqrt{-3}g_{-13}$
matching e: $g_{13}+\left(1/2\sqrt{-3}-1/2\right)g_{6}+\left(-1/2\sqrt{-3}-1/2\right)g_{5}+\left(1/2\sqrt{-3}+1/2\right)g_{-5}+\left(-1/2\sqrt{-3}+1/2\right)g_{-6}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 2/3\sqrt{-3} & -2/3\sqrt{-3}\\ -2/3\sqrt{-3} & 0 & 2/3\sqrt{-3}\\ 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2/3\sqrt{-3}g_{13}-2/3\sqrt{-3}g_{6}-2/3\sqrt{-3}g_{5}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-6}+2/3\sqrt{-3}g_{-13}$
matching e: $g_{13}+\left(1/2\sqrt{-3}-1/2\right)g_{6}+\left(-1/2\sqrt{-3}-1/2\right)g_{5}+\left(1/2\sqrt{-3}+1/2\right)g_{-5}+\left(-1/2\sqrt{-3}+1/2\right)g_{-6}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 2/3\sqrt{-3} & -2/3\sqrt{-3}\\ -2/3\sqrt{-3} & 0 & 2/3\sqrt{-3}\\ 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}288\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{48}_1$
h-characteristic: (0, 2, 0, 0, 0, 2, 0, 0)
Length of the weight dual to h: 96
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{2}_4+A^{2}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{8\omega_{1}}+5V_{6\omega_{1}}+10V_{4\omega_{1}}+6V_{2\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&16h_{8}+32h_{7}+32h_{6}+28h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&2g_{32}+4g_{29}+6g_{28}+6g_{25}+2g_{20}+4g_{2}\\ f&=&g_{-2}+g_{-20}+g_{-25}+g_{-28}+g_{-29}+g_{-32}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&16h_{8}+32h_{7}+32h_{6}+28h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ [h, e]&=&4g_{32}+8g_{29}+12g_{28}+12g_{25}+4g_{20}+8g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-20}-2g_{-25}-2g_{-28}-2g_{-29}-2g_{-32}\end{array}$
Centralizer type: $A^{5}_1+A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 6): $h_{7}+h_{5}+2h_{4}+h_{3}-h_{1}$, $h_{8}-2h_{4}$, $g_{1}+g_{-18}+g_{-22}$, $g_{4}+g_{-8}$, $g_{8}+g_{-4}$, $g_{22}+g_{18}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 2): $1/2h_{8}-h_{4}$, $h_{8}+h_{7}+h_{5}+h_{3}-h_{1}$
Cartan of centralizer (dimension: 2): $1/2h_{8}-h_{4}$, $h_{8}+h_{7}+h_{5}+h_{3}-h_{1}$
Cartan-generating semisimple element: $3/2h_{8}+h_{7}+h_{5}-h_{4}+h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6-5x^4+4x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -2)(x -1)(x +1)(x +2)
Eigenvalues of ad H: $0$, $2$, $1$, $-1$, $-2$
6 eigenvectors of ad H: $(1,0,0,0,0,0)$, $(0,1,0,0,0,0)$, $(0,0,0,0,1,0)$, $(0,0,0,0,0,1)$, $(0,0,1,0,0,0)$, $(0,0,0,1,0,0)$
Centralizer type: A^{5}_1+A^{3}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}-2h_{4}$
matching e: $g_{8}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}-2h_{4}$
matching e: $g_{8}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}+2h_{5}+2h_{4}+2h_{3}-2h_{1}$
matching e: $g_{22}+g_{18}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}+2h_{5}+2h_{4}+2h_{3}-2h_{1}$
matching e: $g_{22}+g_{18}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{47}_1$
h-characteristic: (2, 0, 2, 0, 0, 0, 1, 0)
Length of the weight dual to h: 94
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 8
Containing regular semisimple subalgebra number 1: $C^{1}_3+B^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_3+B^{1}_2+A^{2}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_6+A^{1}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_5+2A^{1}_1$ Containing regular semisimple subalgebra number 5: $C^{1}_4+C^{1}_3$ Containing regular semisimple subalgebra number 6: $C^{1}_4+B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 7: $2C^{1}_3+A^{1}_1$ Containing regular semisimple subalgebra number 8: $C^{1}_5+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+V_{8\omega_{1}}+5V_{6\omega_{1}}+2V_{5\omega_{1}}+5V_{4\omega_{1}}+2V_{3\omega_{1}}+8V_{2\omega_{1}}+4V_{\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&15h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&4g_{52}+g_{44}+g_{34}+8g_{25}+9g_{22}+3g_{10}+5g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-22}+g_{-25}+g_{-34}+g_{-44}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&15h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&8g_{52}+2g_{44}+2g_{34}+16g_{25}+18g_{22}+6g_{10}+10g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-22}-2g_{-25}-2g_{-34}-2g_{-44}-2g_{-52}\end{array}$
Centralizer type: $A_1$
Unfold the hidden panel for more information.

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$A^{46}_1$
h-characteristic: (2, 0, 2, 0, 0, 1, 0, 0)
Length of the weight dual to h: 92
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $C^{1}_3+B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_6$ Containing regular semisimple subalgebra number 3: $C^{1}_5+A^{1}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_4+B^{1}_2$ Containing regular semisimple subalgebra number 5: $2C^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+V_{8\omega_{1}}+4V_{6\omega_{1}}+4V_{5\omega_{1}}+3V_{4\omega_{1}}+4V_{3\omega_{1}}+5V_{2\omega_{1}}+4V_{\omega_{1}}+10V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&14h_{8}+28h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&4g_{52}+g_{44}+9g_{34}+8g_{18}+3g_{10}+5g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-18}+g_{-34}+g_{-44}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&14h_{8}+28h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&8g_{52}+2g_{44}+18g_{34}+16g_{18}+6g_{10}+10g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-18}-2g_{-34}-2g_{-44}-2g_{-52}\end{array}$
Centralizer type: $B_2$
Unfold the hidden panel for more information.

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$A^{45}_1$
h-characteristic: (2, 1, 0, 1, 0, 0, 1, 0)
Length of the weight dual to h: 90
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $C^{1}_3+A^{2}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_3+A^{2}_2+A^{2}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_4+A^{2}_2+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+2V_{7\omega_{1}}+3V_{6\omega_{1}}+2V_{5\omega_{1}}+5V_{4\omega_{1}}+6V_{3\omega_{1}}+5V_{2\omega_{1}}+4V_{\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&15h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ e&=&g_{44}+2g_{37}+g_{34}+8g_{30}+2g_{26}+9g_{22}+5g_{1}\\ f&=&g_{-1}+g_{-22}+g_{-26}+g_{-30}+g_{-34}+g_{-37}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&15h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&2g_{44}+4g_{37}+2g_{34}+16g_{30}+4g_{26}+18g_{22}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-22}-2g_{-26}-2g_{-30}-2g_{-34}-2g_{-37}-2g_{-44}\end{array}$
Centralizer type: $A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 4): $h_{8}-2h_{3}$, $g_{3}+g_{-8}$, $g_{5}-g_{-5}$, $g_{8}+g_{-3}$
Basis of centralizer intersected with cartan (dimension: 1): $1/2h_{8}-h_{3}$
Cartan of centralizer (dimension: 2): $1/2h_{8}-h_{3}$, $g_{5}-g_{-5}$
Cartan-generating semisimple element: $1/2h_{8}-h_{3}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^4-x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -1)(x +1)
Eigenvalues of ad H: $0$, $1$, $-1$
4 eigenvectors of ad H: $(1,0,0,0)$, $(0,0,1,0)$, $(0,0,0,1)$, $(0,1,0,0)$
Centralizer type: A^{3}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}-2h_{3}$
matching e: $g_{8}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}-2h_{3}$
matching e: $g_{8}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{45}_1$
h-characteristic: (2, 0, 2, 0, 1, 0, 0, 0)
Length of the weight dual to h: 90
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_3+B^{1}_2$ Containing regular semisimple subalgebra number 2: $C^{1}_5$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+V_{8\omega_{1}}+3V_{6\omega_{1}}+6V_{5\omega_{1}}+V_{4\omega_{1}}+6V_{3\omega_{1}}+3V_{2\omega_{1}}+21V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&13h_{8}+26h_{7}+26h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&4g_{52}+9g_{44}+8g_{11}+3g_{10}+5g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-11}+g_{-44}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&13h_{8}+26h_{7}+26h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&8g_{52}+18g_{44}+16g_{11}+6g_{10}+10g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-11}-2g_{-44}-2g_{-52}\end{array}$
Centralizer type: $C_3$
Unfold the hidden panel for more information.

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$A^{44}_1$
h-characteristic: (2, 1, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 88
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_3+A^{2}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_4+A^{2}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+2V_{7\omega_{1}}+2V_{6\omega_{1}}+4V_{5\omega_{1}}+4V_{4\omega_{1}}+4V_{3\omega_{1}}+7V_{2\omega_{1}}+4V_{\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&14h_{8}+28h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ e&=&g_{44}+2g_{42}+9g_{34}+8g_{24}+2g_{19}+5g_{1}\\ f&=&g_{-1}+g_{-19}+g_{-24}+g_{-34}+g_{-42}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&14h_{8}+28h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&2g_{44}+4g_{42}+18g_{34}+16g_{24}+4g_{19}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-19}-2g_{-24}-2g_{-34}-2g_{-42}-2g_{-44}\end{array}$
Centralizer type: $A^{3}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 6): $g_{-8}$, $h_{7}-h_{3}$, $h_{8}$, $g_{3}+g_{-22}$, $g_{8}$, $g_{22}+g_{-3}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{7}-h_{3}$, $-h_{8}$
Cartan of centralizer (dimension: 2): $h_{7}-h_{3}$, $-h_{8}$
Cartan-generating semisimple element: $-h_{8}+h_{7}-h_{3}$
adjoint action: $\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6-10x^4+9x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -3)(x -1)(x +1)(x +3)
Eigenvalues of ad H: $0$, $3$, $1$, $-1$, $-3$
6 eigenvectors of ad H: $(0,1,0,0,0,0)$, $(0,0,1,0,0,0)$, $(1,0,0,0,0,0)$, $(0,0,0,0,0,1)$, $(0,0,0,1,0,0)$, $(0,0,0,0,1,0)$
Centralizer type: A^{3}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}-2h_{3}$
matching e: $g_{22}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}-2h_{3}$
matching e: $g_{22}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{43}_1$
h-characteristic: (2, 1, 0, 1, 1, 0, 0, 0)
Length of the weight dual to h: 86
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $C^{1}_3+A^{2}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+2V_{7\omega_{1}}+V_{6\omega_{1}}+6V_{5\omega_{1}}+3V_{4\omega_{1}}+2V_{3\omega_{1}}+10V_{2\omega_{1}}+13V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&13h_{8}+26h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ e&=&2g_{47}+9g_{44}+8g_{17}+2g_{12}+5g_{1}\\ f&=&g_{-1}+g_{-12}+g_{-17}+g_{-44}+g_{-47}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&13h_{8}+26h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&4g_{47}+18g_{44}+16g_{17}+4g_{12}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-12}-2g_{-17}-2g_{-44}-2g_{-47}\end{array}$
Centralizer type: $B_2+A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 13): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $h_{6}-h_{3}$, $h_{7}$, $h_{8}$, $g_{3}+g_{-34}$, $g_{7}$, $g_{8}$, $g_{15}$, $g_{22}$, $g_{34}+g_{-3}$
Basis of centralizer intersected with cartan (dimension: 3): $h_{6}-h_{3}$, $-h_{8}$, $-h_{7}$
Cartan of centralizer (dimension: 3): $-h_{8}$, $h_{6}-h_{3}$, $-h_{7}$
Cartan-generating semisimple element: $-h_{8}-h_{7}+h_{6}-h_{3}$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{13}-17/2x^{11}+393/16x^9-239/8x^7+241/16x^5-9/4x^3$
Factorization of characteristic polynomial of ad H: 1/16(x )(x )(x )(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(2x -3)(2x -1)(2x +1)(2x +3)
Eigenvalues of ad H: $0$, $2$, $1$, $-1$, $-2$, $3/2$, $1/2$, $-1/2$, $-3/2$
13 eigenvectors of ad H: $(0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0)$
Centralizer type: B^{1}_2+A^{3}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9 & -1/18\\ -1/18 & 1/18\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 0), (-1, 0), (2, 1), (-2, -1), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36 & -36\\ -36 & 72\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}+2h_{6}-2h_{3}$
matching e: $g_{34}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}+2h_{6}-2h_{3}$
matching e: $g_{34}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{43}_1$
h-characteristic: (0, 2, 0, 1, 0, 0, 1, 0)
Length of the weight dual to h: 86
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{2}_4+3A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_4+A^{2}_1+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{8\omega_{1}}+V_{6\omega_{1}}+6V_{5\omega_{1}}+3V_{4\omega_{1}}+6V_{3\omega_{1}}+7V_{2\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&15h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&g_{44}+g_{34}+6g_{32}+6g_{31}+g_{22}+4g_{16}+4g_{2}\\ f&=&g_{-2}+g_{-16}+g_{-22}+g_{-31}+g_{-32}+g_{-34}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&15h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ [h, e]&=&2g_{44}+2g_{34}+12g_{32}+12g_{31}+2g_{22}+8g_{16}+8g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-16}-2g_{-22}-2g_{-31}-2g_{-32}-2g_{-34}-2g_{-44}\end{array}$
Centralizer type: $A^{8}_1+A^{5}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 6): $h_{8}+2h_{3}-2h_{1}$, $g_{1}+g_{-3}+g_{-8}$, $g_{5}-g_{-5}$, $g_{6}-g_{-6}$, $g_{8}+g_{3}+g_{-1}$, $g_{13}-g_{-13}$
Basis of centralizer intersected with cartan (dimension: 1): $1/2h_{8}+h_{3}-h_{1}$
Cartan of centralizer (dimension: 2): $1/2g_{8}+1/2g_{3}+1/2g_{1}+1/2h_{8}+h_{3}-h_{1}+1/2g_{-1}+1/2g_{-3}+1/2g_{-8}$, $g_{13}+g_{6}+g_{5}-g_{-5}-g_{-6}-g_{-13}$
Cartan-generating semisimple element: $g_{13}+g_{8}+g_{6}+g_{5}+g_{3}+g_{1}+h_{8}+2h_{3}-2h_{1}+g_{-1}+g_{-3}-g_{-5}-g_{-6}+g_{-8}-g_{-13}$
adjoint action: $\begin{pmatrix}0 & 1 & 0 & 0 & -1 & 0\\ 2 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & -1\\ -2 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & -1 & 1 & 0 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6-5x^4-24x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x^2-8)(x^2+3)
Eigenvalues of ad H: $0$, $2\sqrt{2}$, $-2\sqrt{2}$, $\sqrt{-3}$, $-\sqrt{-3}$
6 eigenvectors of ad H: $(1,1,0,0,1,0)$, $(0,0,1,1,0,1)$, $(−2+1,22−3,0,0,1,0)$, $(2+1,−22−3,0,0,1,0)$, $(0,0,−12−3−12,12−3−12,0,1)$, $(0,0,12−3−12,−12−3−12,0,1)$
Centralizer type: A^{8}_1+A^{5}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$1/2\sqrt{2}g_{8}+1/2\sqrt{2}g_{3}+1/2\sqrt{2}g_{1}+1/2\sqrt{2}h_{8}+\sqrt{2}h_{3}-\sqrt{2}h_{1}+1/2\sqrt{2}g_{-1}+1/2\sqrt{2}g_{-3}+1/2\sqrt{2}g_{-8}$
matching e: $g_{8}+g_{3}+\left(2\sqrt{2}-3\right)g_{1}+\left(-\sqrt{2}+1\right)h_{8}+\left(-2\sqrt{2}+2\right)h_{3}+\left(2\sqrt{2}-2\right)h_{1}+g_{-1}+\left(2\sqrt{2}-3\right)g_{-3}+\left(2\sqrt{2}-3\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 1/2\sqrt{2} & 0 & 0 & -1/2\sqrt{2} & 0\\ \sqrt{2} & -\sqrt{2} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ -\sqrt{2} & 0 & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$1/2\sqrt{2}g_{8}+1/2\sqrt{2}g_{3}+1/2\sqrt{2}g_{1}+1/2\sqrt{2}h_{8}+\sqrt{2}h_{3}-\sqrt{2}h_{1}+1/2\sqrt{2}g_{-1}+1/2\sqrt{2}g_{-3}+1/2\sqrt{2}g_{-8}$
matching e: $g_{8}+g_{3}+\left(2\sqrt{2}-3\right)g_{1}+\left(-\sqrt{2}+1\right)h_{8}+\left(-2\sqrt{2}+2\right)h_{3}+\left(2\sqrt{2}-2\right)h_{1}+g_{-1}+\left(2\sqrt{2}-3\right)g_{-3}+\left(2\sqrt{2}-3\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 1/2\sqrt{2} & 0 & 0 & -1/2\sqrt{2} & 0\\ \sqrt{2} & -\sqrt{2} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ -\sqrt{2} & 0 & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2/3\sqrt{-3}g_{13}-2/3\sqrt{-3}g_{6}-2/3\sqrt{-3}g_{5}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-6}+2/3\sqrt{-3}g_{-13}$
matching e: $g_{13}+\left(1/2\sqrt{-3}-1/2\right)g_{6}+\left(-1/2\sqrt{-3}-1/2\right)g_{5}+\left(1/2\sqrt{-3}+1/2\right)g_{-5}+\left(-1/2\sqrt{-3}+1/2\right)g_{-6}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2/3\sqrt{-3}g_{13}-2/3\sqrt{-3}g_{6}-2/3\sqrt{-3}g_{5}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-6}+2/3\sqrt{-3}g_{-13}$
matching e: $g_{13}+\left(1/2\sqrt{-3}-1/2\right)g_{6}+\left(-1/2\sqrt{-3}-1/2\right)g_{5}+\left(1/2\sqrt{-3}+1/2\right)g_{-5}+\left(-1/2\sqrt{-3}+1/2\right)g_{-6}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}288\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{42}_1$
h-characteristic: (0, 2, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 84
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{2}_4+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_4+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{8\omega_{1}}+V_{6\omega_{1}}+4V_{5\omega_{1}}+7V_{4\omega_{1}}+4V_{3\omega_{1}}+4V_{2\omega_{1}}+4V_{\omega_{1}}+7V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&14h_{8}+28h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&g_{44}+6g_{38}+g_{34}+6g_{25}+4g_{16}+4g_{2}\\ f&=&g_{-2}+g_{-16}+g_{-25}+g_{-34}+g_{-38}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&14h_{8}+28h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ [h, e]&=&2g_{44}+12g_{38}+2g_{34}+12g_{25}+8g_{16}+8g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-16}-2g_{-25}-2g_{-34}-2g_{-38}-2g_{-44}\end{array}$
Centralizer type: $A^{5}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 7): $g_{-8}$, $h_{7}+h_{3}-h_{1}$, $h_{8}$, $g_{1}+g_{-3}+g_{-22}$, $g_{5}-g_{-5}$, $g_{8}$, $g_{22}+g_{3}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{7}+h_{3}-h_{1}$, $-h_{8}$
Cartan of centralizer (dimension: 3): $g_{5}-g_{-5}$, $-h_{8}$, $h_{7}+h_{3}-h_{1}$
Cartan-generating semisimple element: $-h_{8}+h_{7}+h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^7-10x^5+9x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -3)(x -1)(x +1)(x +3)
Eigenvalues of ad H: $0$, $3$, $1$, $-1$, $-3$
7 eigenvectors of ad H: $(0,1,0,0,0,0,0)$, $(0,0,1,0,0,0,0)$, $(0,0,0,0,1,0,0)$, $(1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1)$, $(0,0,0,1,0,0,0)$, $(0,0,0,0,0,1,0)$
Centralizer type: A^{5}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}+2h_{3}-2h_{1}$
matching e: $g_{22}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}+2h_{3}-2h_{1}$
matching e: $g_{22}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{41}_1$
h-characteristic: (0, 2, 0, 1, 1, 0, 0, 0)
Length of the weight dual to h: 82
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{2}_4+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{8\omega_{1}}+V_{6\omega_{1}}+2V_{5\omega_{1}}+11V_{4\omega_{1}}+2V_{3\omega_{1}}+2V_{2\omega_{1}}+4V_{\omega_{1}}+13V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&13h_{8}+26h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&g_{44}+6g_{43}+6g_{18}+4g_{16}+4g_{2}\\ f&=&g_{-2}+g_{-16}+g_{-18}+g_{-43}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&13h_{8}+26h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ [h, e]&=&2g_{44}+12g_{43}+12g_{18}+8g_{16}+8g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-16}-2g_{-18}-2g_{-43}-2g_{-44}\end{array}$
Centralizer type: $B_2+A^{5}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 13): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $h_{6}+h_{3}-h_{1}$, $h_{7}$, $h_{8}$, $g_{1}+g_{-3}+g_{-34}$, $g_{7}$, $g_{8}$, $g_{15}$, $g_{22}$, $g_{34}+g_{3}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 3): $h_{6}+h_{3}-h_{1}$, $-h_{8}$, $-h_{7}$
Cartan of centralizer (dimension: 3): $-h_{7}$, $-h_{8}$, $h_{6}+h_{3}-h_{1}$
Cartan-generating semisimple element: $-h_{8}-h_{7}+h_{6}+h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{13}-17/2x^{11}+393/16x^9-239/8x^7+241/16x^5-9/4x^3$
Factorization of characteristic polynomial of ad H: 1/16(x )(x )(x )(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(2x -3)(2x -1)(2x +1)(2x +3)
Eigenvalues of ad H: $0$, $2$, $1$, $-1$, $-2$, $3/2$, $1/2$, $-1/2$, $-3/2$
13 eigenvectors of ad H: $(0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0)$
Centralizer type: B^{1}_2+A^{5}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9 & -1/18\\ -1/18 & 1/18\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 0), (-1, 0), (2, 1), (-2, -1), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36 & -36\\ -36 & 72\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}+2h_{6}+2h_{3}-2h_{1}$
matching e: $g_{34}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}+2h_{6}+2h_{3}-2h_{1}$
matching e: $g_{34}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{40}_1$
h-characteristic: (2, 2, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 80
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 6
Containing regular semisimple subalgebra number 1: $C^{1}_4+4A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_3+5A^{1}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_4+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_3+A^{2}_1+3A^{1}_1$ Containing regular semisimple subalgebra number 5: $C^{1}_4+2A^{2}_1$ Containing regular semisimple subalgebra number 6: $C^{1}_3+2A^{2}_1+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+6V_{6\omega_{1}}+5V_{4\omega_{1}}+16V_{2\omega_{1}}+10V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&14h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&-1/2g_{58}+1/3g_{52}-1/3g_{44}-40/41g_{36}+1/4g_{34}-72/41g_{30}+169/41g_{22}-9/41g_{15}+90/41g_{8}+5g_{1}\\ f&=&g_{-1}+g_{-8}+g_{-15}+2g_{-22}-4g_{-30}+4g_{-34}-g_{-36}-3g_{-44}+3g_{-52}-2g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&14h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&-g_{58}+2/3g_{52}-2/3g_{44}-80/41g_{36}+1/2g_{34}-144/41g_{30}+338/41g_{22}-18/41g_{15}+180/41g_{8}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-8}-2g_{-15}-4g_{-22}+8g_{-30}-8g_{-34}+2g_{-36}+6g_{-44}-6g_{-52}+4g_{-58}\end{array}$
Centralizer type: $B^{4}_2$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 10): $g_{3}+2/3g_{-3}$, $g_{4}+g_{-4}$, $g_{5}+3/4g_{-5}$, $g_{11}-2/3g_{-11}$, $g_{12}-3/4g_{-12}$, $g_{14}-1/4g_{6}+5g_{-6}-9g_{-14}$, $g_{18}+1/2g_{-18}$, $g_{20}-1/4g_{13}-15/4g_{-13}+27/4g_{-20}$, $g_{26}-1/4g_{19}+15/4g_{-19}-27/4g_{-26}$, $g_{31}-1/4g_{25}-5/2g_{-25}+9/2g_{-31}$
Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 2): $-103/75g_{31}-13/9g_{26}+103/300g_{25}-389/225g_{20}+13/36g_{19}-283/75g_{18}-9/5g_{14}+389/900g_{13}-31/9g_{12}+3/10g_{11}+9/20g_{6}-479/225g_{5}-3/2g_{3}-g_{-3}-479/300g_{-5}-9g_{-6}-1/5g_{-11}+31/12g_{-12}+389/60g_{-13}+81/5g_{-14}-283/150g_{-18}-65/12g_{-19}-1167/100g_{-20}+103/30g_{-25}+39/4g_{-26}-309/50g_{-31}$, $-19/225g_{31}-1/27g_{26}+19/900g_{25}+103/675g_{20}+1/108g_{19}+341/225g_{18}+1/5g_{14}-103/2700g_{13}+35/27g_{12}-6/5g_{11}-1/20g_{6}+283/675g_{5}-g_{4}-g_{-4}+283/900g_{-5}+g_{-6}+4/5g_{-11}-35/36g_{-12}-103/180g_{-13}-9/5g_{-14}+341/450g_{-18}-5/36g_{-19}+103/100g_{-20}+19/90g_{-25}+1/4g_{-26}-19/50g_{-31}$
Cartan-generating semisimple element: $g_{31}+g_{26}-1/4g_{25}+g_{20}-1/4g_{19}+g_{18}+g_{14}-1/4g_{13}+g_{12}+g_{11}-1/4g_{6}+g_{5}+g_{4}+g_{3}+2/3g_{-3}+g_{-4}+3/4g_{-5}+5g_{-6}-2/3g_{-11}-3/4g_{-12}-15/4g_{-13}-9g_{-14}+1/2g_{-18}+15/4g_{-19}+27/4g_{-20}-5/2g_{-25}-27/4g_{-26}+9/2g_{-31}$
adjoint action: $\begin{pmatrix}0 & 1 & 0 & -1 & -3/4 & 0 & 3/4 & 0 & -123/16 & 123/16\\ -2/3 & 0 & 3/4 & 2/3 & -3/4 & 0 & 0 & 123/16 & -123/16 & 0\\ 0 & -1 & 0 & 2/3 & 1 & -41/4 & -2/3 & 41/4 & 0 & 0\\ -1 & 1 & 3/4 & 0 & 0 & 0 & -3/4 & 123/16 & 0 & -123/16\\ -2/3 & -1 & 1 & 0 & 0 & -41/4 & 2/3 & 0 & 41/4 & 0\\ 0 & 0 & -3/4 & 0 & 3/4 & 0 & -1/2 & 3/4 & -3/4 & 1/2\\ -1 & 0 & 1 & -1 & 1 & -41/4 & 0 & 0 & 0 & 41/4\\ 0 & -1 & -1 & 2/3 & 0 & 1 & 0 & 0 & 1 & -2/3\\ -2/3 & -1 & 0 & 0 & -1 & 1 & 0 & 1 & 0 & 2/3\\ -1 & 0 & 0 & -1 & 0 & 1 & -1 & 1 & 1 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{10}+87/8x^8+3411/64x^6+41557/512x^4-74925/512x^2$
Factorization of characteristic polynomial of ad H: 1/512(x )(x )(x -1)(x +1)(8x^2+37)(8x^2-16x +45)(8x^2+16x +45)
Eigenvalues of ad H: $0$, $1$, $-1$, $1/4\sqrt{-74}$, $-1/4\sqrt{-74}$, $1/4\sqrt{-74}+1$, $-1/4\sqrt{-74}+1$, $1/4\sqrt{-74}-1$, $-1/4\sqrt{-74}-1$
10 eigenvectors of ad H: $(5732,−30916,858,−75332,2338,6,1354,5,1,0)$, $(−2532,32516,−778,78532,−2258,−5,−1314,−4,0,1)$, $(−4116,414,−4124,20516,−20524,0,−414,16,56,1)$, $(4564,4516,−1996,31564,−25196,−45164,−378,−341984,611984,1)$, $(−151724536−74−856912268,15176134−74+85693067,348536804−74−25016134,758524536−74+4284512268,−1258736804−74−146376134,1356134−74−22056134,−553512268−74−176716134,1859201−74−13483067,379201−74+21843067,1)$, $(151724536−74−856912268,−15176134−74+85693067,−348536804−74−25016134,−758524536−74+4284512268,1258736804−74−146376134,−1356134−74−22056134,553512268−74−176716134,−1859201−74−13483067,−379201−74+21843067,1)$, $(−116−74−14,14−74+1,−112−74−16,516−74+54,−512−74−56,0,−12−74−1,16,56,1)$, $(116−74−14,−14−74+1,112−74−16,−516−74+54,512−74−56,0,12−74−1,16,56,1)$, $(4564,18−74+418,6112160−74−7792160,31564,−3412160−74−102912160,18−74−12,−378,2512160−74−341540,192160−74+611540,1)$, $(4564,−18−74+418,−6112160−74−7792160,31564,3412160−74−102912160,−18−74−12,−378,−2512160−74−341540,−192160−74+611540,1)$
Centralizer type: B^{4}_2
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/72 & -1/72\\ -1/72 & 1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$-52/1665\sqrt{-74}g_{31}-44/999\sqrt{-74}g_{26}+13/1665\sqrt{-74}g_{25}-476/4995\sqrt{-74}g_{20}+11/999\sqrt{-74}g_{19}-772/1665\sqrt{-74}g_{18}-4/37\sqrt{-74}g_{14}+119/4995\sqrt{-74}g_{13}-404/999\sqrt{-74}g_{12}+10/37\sqrt{-74}g_{11}+1/37\sqrt{-74}g_{6}-836/4995\sqrt{-74}g_{5}+8/37\sqrt{-74}g_{4}-2/37\sqrt{-74}g_{3}-4/111\sqrt{-74}g_{-3}+8/37\sqrt{-74}g_{-4}-209/1665\sqrt{-74}g_{-5}-20/37\sqrt{-74}g_{-6}-20/111\sqrt{-74}g_{-11}+101/333\sqrt{-74}g_{-12}+119/333\sqrt{-74}g_{-13}+36/37\sqrt{-74}g_{-14}-386/1665\sqrt{-74}g_{-18}-55/333\sqrt{-74}g_{-19}-119/185\sqrt{-74}g_{-20}+26/333\sqrt{-74}g_{-25}+11/37\sqrt{-74}g_{-26}-26/185\sqrt{-74}g_{-31}$
matching e: $g_{31}+\left(37/9201\sqrt{-74}+2184/3067\right)g_{26}-1/4g_{25}+\left(185/9201\sqrt{-74}-1348/3067\right)g_{20}+\left(-37/36804\sqrt{-74}-546/3067\right)g_{19}+\left(-5535/12268\sqrt{-74}-17671/6134\right)g_{18}+\left(135/6134\sqrt{-74}-2205/6134\right)g_{14}+\left(-185/36804\sqrt{-74}+337/3067\right)g_{13}+\left(-12587/36804\sqrt{-74}-14637/6134\right)g_{12}+\left(7585/24536\sqrt{-74}+42845/12268\right)g_{11}+\left(-135/24536\sqrt{-74}+2205/24536\right)g_{6}+\left(3485/36804\sqrt{-74}-2501/6134\right)g_{5}+\left(1517/6134\sqrt{-74}+8569/3067\right)g_{4}+\left(-1517/24536\sqrt{-74}-8569/12268\right)g_{3}+\left(-1517/36804\sqrt{-74}-8569/18402\right)g_{-3}+\left(1517/6134\sqrt{-74}+8569/3067\right)g_{-4}+\left(3485/49072\sqrt{-74}-7503/24536\right)g_{-5}+\left(675/6134\sqrt{-74}-11025/6134\right)g_{-6}+\left(-7585/36804\sqrt{-74}-42845/18402\right)g_{-11}+\left(12587/49072\sqrt{-74}+43911/24536\right)g_{-12}+\left(-925/12268\sqrt{-74}+5055/3067\right)g_{-13}+\left(-1215/6134\sqrt{-74}+19845/6134\right)g_{-14}+\left(-5535/24536\sqrt{-74}-17671/12268\right)g_{-18}+\left(185/12268\sqrt{-74}+8190/3067\right)g_{-19}+\left(1665/12268\sqrt{-74}-9099/3067\right)g_{-20}-5/2g_{-25}+\left(-333/12268\sqrt{-74}-14742/3067\right)g_{-26}+9/2g_{-31}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 10/37\sqrt{-74} & 0 & -8/37\sqrt{-74} & 193/555\sqrt{-74} & 0 & -101/333\sqrt{-74} & 0 & 533/2220\sqrt{-74} & -451/1332\sqrt{-74}\\ -20/111\sqrt{-74} & 0 & -101/333\sqrt{-74} & -4/111\sqrt{-74} & 209/1665\sqrt{-74} & 0 & 0 & -451/1332\sqrt{-74} & 4879/6660\sqrt{-74} & 0\\ 0 & 404/999\sqrt{-74} & 0 & -1544/4995\sqrt{-74} & 8/37\sqrt{-74} & 4879/4995\sqrt{-74} & -20/111\sqrt{-74} & -41/37\sqrt{-74} & 0 & 0\\ -8/37\sqrt{-74} & -2/37\sqrt{-74} & -193/555\sqrt{-74} & 0 & 0 & 0 & 209/1665\sqrt{-74} & -533/2220\sqrt{-74} & 0 & 4879/6660\sqrt{-74}\\ 1544/4995\sqrt{-74} & 836/4995\sqrt{-74} & 8/37\sqrt{-74} & 0 & 0 & 451/999\sqrt{-74} & -4/111\sqrt{-74} & 0 & -41/37\sqrt{-74} & 0\\ 0 & 0 & 119/1665\sqrt{-74} & 0 & -11/333\sqrt{-74} & 0 & 26/1665\sqrt{-74} & -209/1665\sqrt{-74} & 101/333\sqrt{-74} & -386/1665\sqrt{-74}\\ 404/999\sqrt{-74} & 0 & 10/37\sqrt{-74} & 836/4995\sqrt{-74} & -2/37\sqrt{-74} & 533/1665\sqrt{-74} & 0 & 0 & 0 & -41/37\sqrt{-74}\\ 0 & 44/999\sqrt{-74} & 4/37\sqrt{-74} & -104/4995\sqrt{-74} & 0 & -836/4995\sqrt{-74} & 0 & 0 & 8/37\sqrt{-74} & -20/111\sqrt{-74}\\ 104/4995\sqrt{-74} & 476/4995\sqrt{-74} & 0 & 0 & 4/37\sqrt{-74} & -404/999\sqrt{-74} & 0 & 8/37\sqrt{-74} & 0 & -4/111\sqrt{-74}\\ 44/999\sqrt{-74} & 0 & 0 & 476/4995\sqrt{-74} & 0 & -772/1665\sqrt{-74} & 4/37\sqrt{-74} & 10/37\sqrt{-74} & -2/37\sqrt{-74} & 0\\ \end{pmatrix}$
$\left(26/1665\sqrt{-74}+32/45\right)g_{31}+\left(22/999\sqrt{-74}+16/27\right)g_{26}+\left(-13/3330\sqrt{-74}-8/45\right)g_{25}+\left(238/4995\sqrt{-74}+16/135\right)g_{20}+\left(-11/1998\sqrt{-74}-4/27\right)g_{19}+\left(386/1665\sqrt{-74}-148/45\right)g_{18}+2/37\sqrt{-74}g_{14}+\left(-119/9990\sqrt{-74}-4/135\right)g_{13}+\left(202/999\sqrt{-74}-74/27\right)g_{12}+\left(-5/37\sqrt{-74}+7/2\right)g_{11}-1/74\sqrt{-74}g_{6}+\left(418/4995\sqrt{-74}-74/135\right)g_{5}+\left(-4/37\sqrt{-74}+3\right)g_{4}+\left(1/37\sqrt{-74}+1/2\right)g_{3}+\left(2/111\sqrt{-74}+1/3\right)g_{-3}+\left(-4/37\sqrt{-74}+3\right)g_{-4}+\left(209/3330\sqrt{-74}-37/90\right)g_{-5}+10/37\sqrt{-74}g_{-6}+\left(10/111\sqrt{-74}-7/3\right)g_{-11}+\left(-101/666\sqrt{-74}+37/18\right)g_{-12}+\left(-119/666\sqrt{-74}-4/9\right)g_{-13}-18/37\sqrt{-74}g_{-14}+\left(193/1665\sqrt{-74}-74/45\right)g_{-18}+\left(55/666\sqrt{-74}+20/9\right)g_{-19}+\left(119/370\sqrt{-74}+4/5\right)g_{-20}+\left(-13/333\sqrt{-74}-16/9\right)g_{-25}+\left(-11/74\sqrt{-74}-4\right)g_{-26}+\left(13/185\sqrt{-74}+16/5\right)g_{-31}$
matching e: $g_{31}+5/6g_{26}-1/4g_{25}+1/6g_{20}-5/24g_{19}+\left(1/2\sqrt{-74}-1\right)g_{18}-1/24g_{13}+\left(5/12\sqrt{-74}-5/6\right)g_{12}+\left(-5/16\sqrt{-74}+5/4\right)g_{11}+\left(1/12\sqrt{-74}-1/6\right)g_{5}+\left(-1/4\sqrt{-74}+1\right)g_{4}+\left(1/16\sqrt{-74}-1/4\right)g_{3}+\left(1/24\sqrt{-74}-1/6\right)g_{-3}+\left(-1/4\sqrt{-74}+1\right)g_{-4}+\left(1/16\sqrt{-74}-1/8\right)g_{-5}+\left(5/24\sqrt{-74}-5/6\right)g_{-11}+\left(-5/16\sqrt{-74}+5/8\right)g_{-12}-5/8g_{-13}+\left(1/4\sqrt{-74}-1/2\right)g_{-18}+25/8g_{-19}+9/8g_{-20}-5/2g_{-25}-45/8g_{-26}+9/2g_{-31}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & -5/37\sqrt{-74}+7/2 & 0 & 4/37\sqrt{-74}-3 & -193/1110\sqrt{-74}+37/15 & 0 & 101/666\sqrt{-74}-37/18 & 0 & -533/4440\sqrt{-74}-82/15 & 451/2664\sqrt{-74}+41/9\\ 10/111\sqrt{-74}-7/3 & 0 & 101/666\sqrt{-74}-37/18 & 2/111\sqrt{-74}+1/3 & -209/3330\sqrt{-74}+37/90 & 0 & 0 & 451/2664\sqrt{-74}+41/9 & -4879/13320\sqrt{-74}-41/45 & 0\\ 0 & -202/999\sqrt{-74}+74/27 & 0 & 772/4995\sqrt{-74}-296/135 & -4/37\sqrt{-74}+3 & -4879/9990\sqrt{-74}-164/135 & 10/111\sqrt{-74}-7/3 & 41/74\sqrt{-74} & 0 & 0\\ 4/37\sqrt{-74}-3 & 1/37\sqrt{-74}+1/2 & 193/1110\sqrt{-74}-37/15 & 0 & 0 & 0 & -209/3330\sqrt{-74}+37/90 & 533/4440\sqrt{-74}+82/15 & 0 & -4879/13320\sqrt{-74}-41/45\\ -772/4995\sqrt{-74}+296/135 & -418/4995\sqrt{-74}+74/135 & -4/37\sqrt{-74}+3 & 0 & 0 & -451/1998\sqrt{-74}-164/27 & 2/111\sqrt{-74}+1/3 & 0 & 41/74\sqrt{-74} & 0\\ 0 & 0 & -119/3330\sqrt{-74}-4/45 & 0 & 11/666\sqrt{-74}+4/9 & 0 & -13/1665\sqrt{-74}-16/45 & 209/3330\sqrt{-74}-37/90 & -101/666\sqrt{-74}+37/18 & 193/1665\sqrt{-74}-74/45\\ -202/999\sqrt{-74}+74/27 & 0 & -5/37\sqrt{-74}+7/2 & -418/4995\sqrt{-74}+74/135 & 1/37\sqrt{-74}+1/2 & -533/3330\sqrt{-74}-328/45 & 0 & 0 & 0 & 41/74\sqrt{-74}\\ 0 & -22/999\sqrt{-74}-16/27 & -2/37\sqrt{-74} & 52/4995\sqrt{-74}+64/135 & 0 & 418/4995\sqrt{-74}-74/135 & 0 & 0 & -4/37\sqrt{-74}+3 & 10/111\sqrt{-74}-7/3\\ -52/4995\sqrt{-74}-64/135 & -238/4995\sqrt{-74}-16/135 & 0 & 0 & -2/37\sqrt{-74} & 202/999\sqrt{-74}-74/27 & 0 & -4/37\sqrt{-74}+3 & 0 & 2/111\sqrt{-74}+1/3\\ -22/999\sqrt{-74}-16/27 & 0 & 0 & -238/4995\sqrt{-74}-16/135 & 0 & 386/1665\sqrt{-74}-148/45 & -2/37\sqrt{-74} & -5/37\sqrt{-74}+7/2 & 1/37\sqrt{-74}+1/2 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-52/1665\sqrt{-74}g_{31}-44/999\sqrt{-74}g_{26}+13/1665\sqrt{-74}g_{25}-476/4995\sqrt{-74}g_{20}+11/999\sqrt{-74}g_{19}-772/1665\sqrt{-74}g_{18}-4/37\sqrt{-74}g_{14}+119/4995\sqrt{-74}g_{13}-404/999\sqrt{-74}g_{12}+10/37\sqrt{-74}g_{11}+1/37\sqrt{-74}g_{6}-836/4995\sqrt{-74}g_{5}+8/37\sqrt{-74}g_{4}-2/37\sqrt{-74}g_{3}-4/111\sqrt{-74}g_{-3}+8/37\sqrt{-74}g_{-4}-209/1665\sqrt{-74}g_{-5}-20/37\sqrt{-74}g_{-6}-20/111\sqrt{-74}g_{-11}+101/333\sqrt{-74}g_{-12}+119/333\sqrt{-74}g_{-13}+36/37\sqrt{-74}g_{-14}-386/1665\sqrt{-74}g_{-18}-55/333\sqrt{-74}g_{-19}-119/185\sqrt{-74}g_{-20}+26/333\sqrt{-74}g_{-25}+11/37\sqrt{-74}g_{-26}-26/185\sqrt{-74}g_{-31}$
matching e: $g_{31}+\left(37/9201\sqrt{-74}+2184/3067\right)g_{26}-1/4g_{25}+\left(185/9201\sqrt{-74}-1348/3067\right)g_{20}+\left(-37/36804\sqrt{-74}-546/3067\right)g_{19}+\left(-5535/12268\sqrt{-74}-17671/6134\right)g_{18}+\left(135/6134\sqrt{-74}-2205/6134\right)g_{14}+\left(-185/36804\sqrt{-74}+337/3067\right)g_{13}+\left(-12587/36804\sqrt{-74}-14637/6134\right)g_{12}+\left(7585/24536\sqrt{-74}+42845/12268\right)g_{11}+\left(-135/24536\sqrt{-74}+2205/24536\right)g_{6}+\left(3485/36804\sqrt{-74}-2501/6134\right)g_{5}+\left(1517/6134\sqrt{-74}+8569/3067\right)g_{4}+\left(-1517/24536\sqrt{-74}-8569/12268\right)g_{3}+\left(-1517/36804\sqrt{-74}-8569/18402\right)g_{-3}+\left(1517/6134\sqrt{-74}+8569/3067\right)g_{-4}+\left(3485/49072\sqrt{-74}-7503/24536\right)g_{-5}+\left(675/6134\sqrt{-74}-11025/6134\right)g_{-6}+\left(-7585/36804\sqrt{-74}-42845/18402\right)g_{-11}+\left(12587/49072\sqrt{-74}+43911/24536\right)g_{-12}+\left(-925/12268\sqrt{-74}+5055/3067\right)g_{-13}+\left(-1215/6134\sqrt{-74}+19845/6134\right)g_{-14}+\left(-5535/24536\sqrt{-74}-17671/12268\right)g_{-18}+\left(185/12268\sqrt{-74}+8190/3067\right)g_{-19}+\left(1665/12268\sqrt{-74}-9099/3067\right)g_{-20}-5/2g_{-25}+\left(-333/12268\sqrt{-74}-14742/3067\right)g_{-26}+9/2g_{-31}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 10/37\sqrt{-74} & 0 & -8/37\sqrt{-74} & 193/555\sqrt{-74} & 0 & -101/333\sqrt{-74} & 0 & 533/2220\sqrt{-74} & -451/1332\sqrt{-74}\\ -20/111\sqrt{-74} & 0 & -101/333\sqrt{-74} & -4/111\sqrt{-74} & 209/1665\sqrt{-74} & 0 & 0 & -451/1332\sqrt{-74} & 4879/6660\sqrt{-74} & 0\\ 0 & 404/999\sqrt{-74} & 0 & -1544/4995\sqrt{-74} & 8/37\sqrt{-74} & 4879/4995\sqrt{-74} & -20/111\sqrt{-74} & -41/37\sqrt{-74} & 0 & 0\\ -8/37\sqrt{-74} & -2/37\sqrt{-74} & -193/555\sqrt{-74} & 0 & 0 & 0 & 209/1665\sqrt{-74} & -533/2220\sqrt{-74} & 0 & 4879/6660\sqrt{-74}\\ 1544/4995\sqrt{-74} & 836/4995\sqrt{-74} & 8/37\sqrt{-74} & 0 & 0 & 451/999\sqrt{-74} & -4/111\sqrt{-74} & 0 & -41/37\sqrt{-74} & 0\\ 0 & 0 & 119/1665\sqrt{-74} & 0 & -11/333\sqrt{-74} & 0 & 26/1665\sqrt{-74} & -209/1665\sqrt{-74} & 101/333\sqrt{-74} & -386/1665\sqrt{-74}\\ 404/999\sqrt{-74} & 0 & 10/37\sqrt{-74} & 836/4995\sqrt{-74} & -2/37\sqrt{-74} & 533/1665\sqrt{-74} & 0 & 0 & 0 & -41/37\sqrt{-74}\\ 0 & 44/999\sqrt{-74} & 4/37\sqrt{-74} & -104/4995\sqrt{-74} & 0 & -836/4995\sqrt{-74} & 0 & 0 & 8/37\sqrt{-74} & -20/111\sqrt{-74}\\ 104/4995\sqrt{-74} & 476/4995\sqrt{-74} & 0 & 0 & 4/37\sqrt{-74} & -404/999\sqrt{-74} & 0 & 8/37\sqrt{-74} & 0 & -4/111\sqrt{-74}\\ 44/999\sqrt{-74} & 0 & 0 & 476/4995\sqrt{-74} & 0 & -772/1665\sqrt{-74} & 4/37\sqrt{-74} & 10/37\sqrt{-74} & -2/37\sqrt{-74} & 0\\ \end{pmatrix}$
$\left(26/1665\sqrt{-74}+32/45\right)g_{31}+\left(22/999\sqrt{-74}+16/27\right)g_{26}+\left(-13/3330\sqrt{-74}-8/45\right)g_{25}+\left(238/4995\sqrt{-74}+16/135\right)g_{20}+\left(-11/1998\sqrt{-74}-4/27\right)g_{19}+\left(386/1665\sqrt{-74}-148/45\right)g_{18}+2/37\sqrt{-74}g_{14}+\left(-119/9990\sqrt{-74}-4/135\right)g_{13}+\left(202/999\sqrt{-74}-74/27\right)g_{12}+\left(-5/37\sqrt{-74}+7/2\right)g_{11}-1/74\sqrt{-74}g_{6}+\left(418/4995\sqrt{-74}-74/135\right)g_{5}+\left(-4/37\sqrt{-74}+3\right)g_{4}+\left(1/37\sqrt{-74}+1/2\right)g_{3}+\left(2/111\sqrt{-74}+1/3\right)g_{-3}+\left(-4/37\sqrt{-74}+3\right)g_{-4}+\left(209/3330\sqrt{-74}-37/90\right)g_{-5}+10/37\sqrt{-74}g_{-6}+\left(10/111\sqrt{-74}-7/3\right)g_{-11}+\left(-101/666\sqrt{-74}+37/18\right)g_{-12}+\left(-119/666\sqrt{-74}-4/9\right)g_{-13}-18/37\sqrt{-74}g_{-14}+\left(193/1665\sqrt{-74}-74/45\right)g_{-18}+\left(55/666\sqrt{-74}+20/9\right)g_{-19}+\left(119/370\sqrt{-74}+4/5\right)g_{-20}+\left(-13/333\sqrt{-74}-16/9\right)g_{-25}+\left(-11/74\sqrt{-74}-4\right)g_{-26}+\left(13/185\sqrt{-74}+16/5\right)g_{-31}$
matching e: $g_{31}+5/6g_{26}-1/4g_{25}+1/6g_{20}-5/24g_{19}+\left(1/2\sqrt{-74}-1\right)g_{18}-1/24g_{13}+\left(5/12\sqrt{-74}-5/6\right)g_{12}+\left(-5/16\sqrt{-74}+5/4\right)g_{11}+\left(1/12\sqrt{-74}-1/6\right)g_{5}+\left(-1/4\sqrt{-74}+1\right)g_{4}+\left(1/16\sqrt{-74}-1/4\right)g_{3}+\left(1/24\sqrt{-74}-1/6\right)g_{-3}+\left(-1/4\sqrt{-74}+1\right)g_{-4}+\left(1/16\sqrt{-74}-1/8\right)g_{-5}+\left(5/24\sqrt{-74}-5/6\right)g_{-11}+\left(-5/16\sqrt{-74}+5/8\right)g_{-12}-5/8g_{-13}+\left(1/4\sqrt{-74}-1/2\right)g_{-18}+25/8g_{-19}+9/8g_{-20}-5/2g_{-25}-45/8g_{-26}+9/2g_{-31}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & -5/37\sqrt{-74}+7/2 & 0 & 4/37\sqrt{-74}-3 & -193/1110\sqrt{-74}+37/15 & 0 & 101/666\sqrt{-74}-37/18 & 0 & -533/4440\sqrt{-74}-82/15 & 451/2664\sqrt{-74}+41/9\\ 10/111\sqrt{-74}-7/3 & 0 & 101/666\sqrt{-74}-37/18 & 2/111\sqrt{-74}+1/3 & -209/3330\sqrt{-74}+37/90 & 0 & 0 & 451/2664\sqrt{-74}+41/9 & -4879/13320\sqrt{-74}-41/45 & 0\\ 0 & -202/999\sqrt{-74}+74/27 & 0 & 772/4995\sqrt{-74}-296/135 & -4/37\sqrt{-74}+3 & -4879/9990\sqrt{-74}-164/135 & 10/111\sqrt{-74}-7/3 & 41/74\sqrt{-74} & 0 & 0\\ 4/37\sqrt{-74}-3 & 1/37\sqrt{-74}+1/2 & 193/1110\sqrt{-74}-37/15 & 0 & 0 & 0 & -209/3330\sqrt{-74}+37/90 & 533/4440\sqrt{-74}+82/15 & 0 & -4879/13320\sqrt{-74}-41/45\\ -772/4995\sqrt{-74}+296/135 & -418/4995\sqrt{-74}+74/135 & -4/37\sqrt{-74}+3 & 0 & 0 & -451/1998\sqrt{-74}-164/27 & 2/111\sqrt{-74}+1/3 & 0 & 41/74\sqrt{-74} & 0\\ 0 & 0 & -119/3330\sqrt{-74}-4/45 & 0 & 11/666\sqrt{-74}+4/9 & 0 & -13/1665\sqrt{-74}-16/45 & 209/3330\sqrt{-74}-37/90 & -101/666\sqrt{-74}+37/18 & 193/1665\sqrt{-74}-74/45\\ -202/999\sqrt{-74}+74/27 & 0 & -5/37\sqrt{-74}+7/2 & -418/4995\sqrt{-74}+74/135 & 1/37\sqrt{-74}+1/2 & -533/3330\sqrt{-74}-328/45 & 0 & 0 & 0 & 41/74\sqrt{-74}\\ 0 & -22/999\sqrt{-74}-16/27 & -2/37\sqrt{-74} & 52/4995\sqrt{-74}+64/135 & 0 & 418/4995\sqrt{-74}-74/135 & 0 & 0 & -4/37\sqrt{-74}+3 & 10/111\sqrt{-74}-7/3\\ -52/4995\sqrt{-74}-64/135 & -238/4995\sqrt{-74}-16/135 & 0 & 0 & -2/37\sqrt{-74} & 202/999\sqrt{-74}-74/27 & 0 & -4/37\sqrt{-74}+3 & 0 & 2/111\sqrt{-74}+1/3\\ -22/999\sqrt{-74}-16/27 & 0 & 0 & -238/4995\sqrt{-74}-16/135 & 0 & 386/1665\sqrt{-74}-148/45 & -2/37\sqrt{-74} & -5/37\sqrt{-74}+7/2 & 1/37\sqrt{-74}+1/2 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 2), (-1, -2), (1, 0), (-1, 0), (1, 1), (-1, -1), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}288 & -144\\ -144 & 144\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{40}_1$
h-characteristic: (0, 0, 0, 2, 0, 0, 0, 2)
Length of the weight dual to h: 80
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $4B^{1}_2$ Containing regular semisimple subalgebra number 2: $A^{2}_3+2B^{1}_2$ Containing regular semisimple subalgebra number 3: $2A^{2}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $10V_{6\omega_{1}}+6V_{4\omega_{1}}+10V_{2\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&16h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+18h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{44}+4g_{34}+3g_{26}+3g_{25}+3g_{24}+3g_{23}+4g_{22}+4g_{8}\\ f&=&g_{-8}+g_{-22}+g_{-23}+g_{-24}+g_{-25}+g_{-26}+g_{-34}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&16h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+18h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{44}+8g_{34}+6g_{26}+6g_{25}+6g_{24}+6g_{23}+8g_{22}+8g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-22}-2g_{-23}-2g_{-24}-2g_{-25}-2g_{-26}-2g_{-34}-2g_{-44}\end{array}$
Centralizer type: $2A^{8}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 6): $g_{5}+g_{1}-g_{-1}-g_{-5}$, $g_{6}+g_{2}-g_{-2}-g_{-6}$, $g_{7}+g_{3}-g_{-3}-g_{-7}$, $g_{13}+g_{9}-g_{-9}-g_{-13}$, $g_{14}+g_{10}-g_{-10}-g_{-14}$, $g_{20}+g_{16}-g_{-16}-g_{-20}$
Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 2): $g_{20}+g_{16}+g_{7}+g_{6}+g_{5}+g_{3}+g_{2}+g_{1}-g_{-1}-g_{-2}-g_{-3}-g_{-5}-g_{-6}-g_{-7}-g_{-16}-g_{-20}$, $g_{14}+g_{13}+g_{10}+g_{9}-g_{-9}-g_{-10}-g_{-13}-g_{-14}$
Cartan-generating semisimple element: $g_{20}+g_{16}+g_{14}+g_{13}+g_{10}+g_{9}+g_{7}+g_{6}+g_{5}+g_{3}+g_{2}+g_{1}-g_{-1}-g_{-2}-g_{-3}-g_{-5}-g_{-6}-g_{-7}-g_{-9}-g_{-10}-g_{-13}-g_{-14}-g_{-16}-g_{-20}$
adjoint action: $\begin{pmatrix}0 & -1 & 0 & 1 & -1 & 1\\ 1 & 0 & -1 & -1 & 1 & 0\\ 0 & 1 & 0 & 1 & -1 & -1\\ -1 & 1 & -1 & 0 & 0 & 1\\ 1 & -1 & 1 & 0 & 0 & -1\\ -1 & 0 & 1 & -1 & 1 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6+12x^4+32x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x^2+4)(x^2+8)
Eigenvalues of ad H: $0$, $2\sqrt{-1}$, $-2\sqrt{-1}$, $2\sqrt{-2}$, $-2\sqrt{-2}$
6 eigenvectors of ad H: $(0,0,0,1,1,0)$, $(1,1,1,0,0,1)$, $(−−1,−1,−1,0,0,1)$, $(−1,−1,−−1,0,0,1)$, $(−1,1,−1,−−2,−2,1)$, $(−1,1,−1,−2,−−2,1)$
Centralizer type: 2A^{8}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-\sqrt{-1}g_{14}-\sqrt{-1}g_{13}-\sqrt{-1}g_{10}-\sqrt{-1}g_{9}+\sqrt{-1}g_{-9}+\sqrt{-1}g_{-10}+\sqrt{-1}g_{-13}+\sqrt{-1}g_{-14}$
matching e: $g_{20}+g_{16}+\sqrt{-1}g_{7}-g_{6}-\sqrt{-1}g_{5}+\sqrt{-1}g_{3}-g_{2}-\sqrt{-1}g_{1}+\sqrt{-1}g_{-1}+g_{-2}-\sqrt{-1}g_{-3}+\sqrt{-1}g_{-5}+g_{-6}-\sqrt{-1}g_{-7}-g_{-16}-g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & \sqrt{-1} & 0 & 0 & 0 & -\sqrt{-1}\\ -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0\\ 0 & -\sqrt{-1} & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-\sqrt{-1}g_{14}-\sqrt{-1}g_{13}-\sqrt{-1}g_{10}-\sqrt{-1}g_{9}+\sqrt{-1}g_{-9}+\sqrt{-1}g_{-10}+\sqrt{-1}g_{-13}+\sqrt{-1}g_{-14}$
matching e: $g_{20}+g_{16}+\sqrt{-1}g_{7}-g_{6}-\sqrt{-1}g_{5}+\sqrt{-1}g_{3}-g_{2}-\sqrt{-1}g_{1}+\sqrt{-1}g_{-1}+g_{-2}-\sqrt{-1}g_{-3}+\sqrt{-1}g_{-5}+g_{-6}-\sqrt{-1}g_{-7}-g_{-16}-g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & \sqrt{-1} & 0 & 0 & 0 & -\sqrt{-1}\\ -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0\\ 0 & -\sqrt{-1} & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}288\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-1/2\sqrt{-2}g_{20}-1/2\sqrt{-2}g_{16}-1/2\sqrt{-2}g_{7}-1/2\sqrt{-2}g_{6}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{3}-1/2\sqrt{-2}g_{2}-1/2\sqrt{-2}g_{1}+1/2\sqrt{-2}g_{-1}+1/2\sqrt{-2}g_{-2}+1/2\sqrt{-2}g_{-3}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-6}+1/2\sqrt{-2}g_{-7}+1/2\sqrt{-2}g_{-16}+1/2\sqrt{-2}g_{-20}$
matching e: $g_{20}+g_{16}+\sqrt{-2}g_{14}-\sqrt{-2}g_{13}+\sqrt{-2}g_{10}-\sqrt{-2}g_{9}-g_{7}+g_{6}-g_{5}-g_{3}+g_{2}-g_{1}+g_{-1}-g_{-2}+g_{-3}+g_{-5}-g_{-6}+g_{-7}+\sqrt{-2}g_{-9}-\sqrt{-2}g_{-10}+\sqrt{-2}g_{-13}-\sqrt{-2}g_{-14}-g_{-16}-g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & -1/2\sqrt{-2}\\ -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 1/2\sqrt{-2}\\ 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-1/2\sqrt{-2}g_{20}-1/2\sqrt{-2}g_{16}-1/2\sqrt{-2}g_{7}-1/2\sqrt{-2}g_{6}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{3}-1/2\sqrt{-2}g_{2}-1/2\sqrt{-2}g_{1}+1/2\sqrt{-2}g_{-1}+1/2\sqrt{-2}g_{-2}+1/2\sqrt{-2}g_{-3}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-6}+1/2\sqrt{-2}g_{-7}+1/2\sqrt{-2}g_{-16}+1/2\sqrt{-2}g_{-20}$
matching e: $g_{20}+g_{16}+\sqrt{-2}g_{14}-\sqrt{-2}g_{13}+\sqrt{-2}g_{10}-\sqrt{-2}g_{9}-g_{7}+g_{6}-g_{5}-g_{3}+g_{2}-g_{1}+g_{-1}-g_{-2}+g_{-3}+g_{-5}-g_{-6}+g_{-7}+\sqrt{-2}g_{-9}-\sqrt{-2}g_{-10}+\sqrt{-2}g_{-13}-\sqrt{-2}g_{-14}-g_{-16}-g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & -1/2\sqrt{-2}\\ -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 1/2\sqrt{-2}\\ 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}288\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{40}_1$
h-characteristic: (0, 2, 0, 2, 0, 0, 0, 0)
Length of the weight dual to h: 80
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{2}_4$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{8\omega_{1}}+V_{6\omega_{1}}+15V_{4\omega_{1}}+V_{2\omega_{1}}+24V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+24h_{7}+24h_{6}+24h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&6g_{48}+4g_{16}+6g_{11}+4g_{2}\\ f&=&g_{-2}+g_{-11}+g_{-16}+g_{-48}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+24h_{7}+24h_{6}+24h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ [h, e]&=&12g_{48}+8g_{16}+12g_{11}+8g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-11}-2g_{-16}-2g_{-48}\end{array}$
Centralizer type: $C_3+A^{5}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 24): $g_{-34}$, $g_{-28}$, $g_{-22}$, $g_{-21}$, $g_{-15}$, $g_{-14}$, $g_{-8}$, $g_{-7}$, $g_{-6}$, $h_{5}+h_{3}-h_{1}$, $h_{6}$, $h_{7}$, $h_{8}$, $g_{1}+g_{-3}+g_{-44}$, $g_{6}$, $g_{7}$, $g_{8}$, $g_{14}$, $g_{15}$, $g_{21}$, $g_{22}$, $g_{28}$, $g_{34}$, $g_{44}+g_{3}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 4): $h_{5}+h_{3}-h_{1}$, $-h_{8}$, $-h_{7}$, $-h_{6}$
Cartan of centralizer (dimension: 4): $-h_{7}$, $-h_{6}$, $h_{5}+h_{3}-h_{1}$, $-h_{8}$
Cartan-generating semisimple element: $-9h_{8}+7h_{7}+5h_{6}+h_{5}+h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}-4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 21/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 23/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -29/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 25 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -27/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 27/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -25 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 29/2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -23/2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -21/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{24}-1291x^{22}+4674543/8x^{20}-1997462351/16x^{18}+3441686272361/256x^{16}-11049787068411/16x^{14}+1786351718615063/128x^{12}-7476599047022629/64x^{10}+101274541816996881/256x^8-31564288362182949/64x^6+804526423430625/4x^4$
Factorization of characteristic polynomial of ad H: 1/256(x )(x )(x )(x )(x -25)(x -4)(x -3)(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(x +3)(x +4)(x +25)(2x -29)(2x -27)(2x -23)(2x -21)(2x +21)(2x +23)(2x +27)(2x +29)
Eigenvalues of ad H: $0$, $25$, $4$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-4$, $-25$, $29/2$, $27/2$, $23/2$, $21/2$, $-21/2$, $-23/2$, $-27/2$, $-29/2$
24 eigenvectors of ad H: $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: C^{1}_3+A^{5}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/18 & -1/36 & -1/18\\ -1/36 & 1/18 & 0\\ -1/18 & 0 & 1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (3 total):
$2h_{6}$
matching e: $g_{6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{6}$
matching e: $g_{6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (1, 0, 1), (-1, 0, -1), (1, 0, 2), (-1, 0, -2), (0, 0, 1), (0, 0, -1), (1, 0, 0), (-1, 0, 0), (2, 1, 2), (-2, -1, -2), (1, 1, 2), (-1, -1, -2), (1, 1, 0), (-1, -1, 0), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}72 & -36 & -36\\ -36 & 72 & 0\\ -36 & 0 & 36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{3}-2h_{1}$
matching e: $g_{44}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{3}-2h_{1}$
matching e: $g_{44}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{39}_1$
h-characteristic: (2, 2, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 78
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $C^{1}_3+4A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_3+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_3+2A^{2}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_4+3A^{1}_1$ Containing regular semisimple subalgebra number 5: $C^{1}_4+A^{2}_1+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+5V_{6\omega_{1}}+2V_{5\omega_{1}}+4V_{4\omega_{1}}+11V_{2\omega_{1}}+8V_{\omega_{1}}+9V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&13h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&g_{58}+g_{52}+g_{44}+g_{34}+8g_{30}+9g_{22}+5g_{1}\\ f&=&g_{-1}+g_{-22}+g_{-30}+g_{-34}+g_{-44}+g_{-52}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&13h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&2g_{58}+2g_{52}+2g_{44}+2g_{34}+16g_{30}+18g_{22}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-22}-2g_{-30}-2g_{-34}-2g_{-44}-2g_{-52}-2g_{-58}\end{array}$
Centralizer type: $2A^{4}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 9): $g_{-8}$, $h_{8}$, $g_{3}-g_{-3}$, $g_{4}-g_{-4}$, $g_{5}-g_{-5}$, $g_{8}$, $g_{11}-g_{-11}$, $g_{12}-g_{-12}$, $g_{18}-g_{-18}$
Basis of centralizer intersected with cartan (dimension: 1): $-h_{8}$
Cartan of centralizer (dimension: 3): $-g_{8}-h_{8}-g_{-8}$, $g_{12}+g_{11}-g_{-11}-g_{-12}$, $g_{18}+g_{5}+g_{4}+g_{3}-g_{-3}-g_{-4}-g_{-5}-g_{-18}$
Cartan-generating semisimple element: $g_{18}+g_{12}+g_{11}+g_{8}+g_{5}+g_{4}+g_{3}+h_{8}-g_{-3}-g_{-4}-g_{-5}+g_{-8}-g_{-11}-g_{-12}-g_{-18}$
adjoint action: $\begin{pmatrix}-2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 1 & -1 & 1\\ 0 & 0 & 1 & 0 & -1 & 0 & -1 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & -1 & -1\\ 0 & -2 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & -1 & 1 & -1 & 0 & 0 & 0 & 1\\ 0 & 0 & 1 & -1 & 1 & 0 & 0 & 0 & -1\\ 0 & 0 & -1 & 0 & 1 & 0 & -1 & 1 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^9+4x^7-64x^5-256x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x^2-8)(x^2+4)(x^2+8)
Eigenvalues of ad H: $0$, $2\sqrt{2}$, $-2\sqrt{2}$, $2\sqrt{-1}$, $-2\sqrt{-1}$, $2\sqrt{-2}$, $-2\sqrt{-2}$
9 eigenvectors of ad H: $(1,1,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,1,1,0)$, $(0,0,1,1,1,0,0,0,1)$, $(22−3,−2+1,0,0,0,1,0,0,0)$, $(−22−3,2+1,0,0,0,1,0,0,0)$, $(0,0,−−1,−1,−1,0,0,0,1)$, $(0,0,−1,−1,−−1,0,0,0,1)$, $(0,0,−1,1,−1,0,−−2,−2,1)$, $(0,0,−1,1,−1,0,−2,−−2,1)$
Centralizer type: 2A^{4}_1+A^{1}_1
Reductive components (3 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$1/2\sqrt{2}g_{8}+1/2\sqrt{2}h_{8}+1/2\sqrt{2}g_{-8}$
matching e: $g_{8}+\left(-\sqrt{2}+1\right)h_{8}+\left(2\sqrt{2}-3\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & 0 & -1/2\sqrt{2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & 0 & \sqrt{2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$1/2\sqrt{2}g_{8}+1/2\sqrt{2}h_{8}+1/2\sqrt{2}g_{-8}$
matching e: $g_{8}+\left(-\sqrt{2}+1\right)h_{8}+\left(2\sqrt{2}-3\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & 0 & -1/2\sqrt{2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & 0 & \sqrt{2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-\sqrt{-1}g_{12}-\sqrt{-1}g_{11}+\sqrt{-1}g_{-11}+\sqrt{-1}g_{-12}$
matching e: $g_{18}+\sqrt{-1}g_{5}-g_{4}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}+g_{-4}-\sqrt{-1}g_{-5}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & -\sqrt{-1}\\ 0 & 0 & -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-\sqrt{-1}g_{12}-\sqrt{-1}g_{11}+\sqrt{-1}g_{-11}+\sqrt{-1}g_{-12}$
matching e: $g_{18}+\sqrt{-1}g_{5}-g_{4}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}+g_{-4}-\sqrt{-1}g_{-5}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & -\sqrt{-1}\\ 0 & 0 & -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-1/2\sqrt{-2}g_{18}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{4}-1/2\sqrt{-2}g_{3}+1/2\sqrt{-2}g_{-3}+1/2\sqrt{-2}g_{-4}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-18}$
matching e: $g_{18}+\sqrt{-2}g_{12}-\sqrt{-2}g_{11}-g_{5}+g_{4}-g_{3}+g_{-3}-g_{-4}+g_{-5}+\sqrt{-2}g_{-11}-\sqrt{-2}g_{-12}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & 0 & -1/2\sqrt{-2}\\ 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 0 & 1/2\sqrt{-2}\\ 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-1/2\sqrt{-2}g_{18}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{4}-1/2\sqrt{-2}g_{3}+1/2\sqrt{-2}g_{-3}+1/2\sqrt{-2}g_{-4}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-18}$
matching e: $g_{18}+\sqrt{-2}g_{12}-\sqrt{-2}g_{11}-g_{5}+g_{4}-g_{3}+g_{-3}-g_{-4}+g_{-5}+\sqrt{-2}g_{-11}-\sqrt{-2}g_{-12}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & 0 & -1/2\sqrt{-2}\\ 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 0 & 1/2\sqrt{-2}\\ 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{38}_1$
h-characteristic: (2, 2, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 76
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $C^{1}_3+3A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_3+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_4+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_4+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+4V_{6\omega_{1}}+4V_{5\omega_{1}}+3V_{4\omega_{1}}+7V_{2\omega_{1}}+12V_{\omega_{1}}+13V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+24h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&g_{58}+g_{52}+g_{44}+9g_{34}+8g_{24}+5g_{1}\\ f&=&g_{-1}+g_{-24}+g_{-34}+g_{-44}+g_{-52}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+24h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&2g_{58}+2g_{52}+2g_{44}+18g_{34}+16g_{24}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-24}-2g_{-34}-2g_{-44}-2g_{-52}-2g_{-58}\end{array}$
Centralizer type: $B_2+A^{8}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 13): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $h_{7}$, $h_{8}$, $g_{3}-g_{-3}$, $g_{4}-g_{-4}$, $g_{7}$, $g_{8}$, $g_{11}-g_{-11}$, $g_{15}$, $g_{22}$
Basis of centralizer intersected with cartan (dimension: 2): $-h_{8}$, $-h_{7}$
Cartan of centralizer (dimension: 3): $-g_{22}-g_{15}-g_{8}-3h_{8}-3h_{7}-g_{-8}-g_{-15}-g_{-22}$, $g_{11}+g_{4}+g_{3}-g_{-3}-g_{-4}-g_{-11}$, $-g_{7}+2h_{8}+2h_{7}-g_{-7}$
Cartan-generating semisimple element: $g_{22}+g_{15}+g_{11}+g_{8}+g_{7}+g_{4}+g_{3}+h_{8}+h_{7}-g_{-3}-g_{-4}+g_{-7}+g_{-8}-g_{-11}+g_{-15}+g_{-22}$
adjoint action: $\begin{pmatrix}-1 & -2 & 0 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & -1 & -1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & -2 & -1 & 0 & -1 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & -1 & 0 & -1 & 0\\ 2 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & -2 & -2\\ 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -2 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 1 & 1 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & -1\\ 0 & 0 & 0 & -2 & 1 & -2 & 0 & 0 & 0 & 1 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & -1 & 0 & 0 & -1 & 1 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -2 & 0 & 0 & 2 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{13}-36x^{11}+282x^9-64x^7-2883x^5+2700x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -5)(x -3)(x -2)(x -1)(x +1)(x +2)(x +3)(x +5)(x^2+3)
Eigenvalues of ad H: $0$, $5$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-5$, $\sqrt{-3}$, $-\sqrt{-3}$
13 eigenvectors of ad H: $(0,0,0,1,−2,−2,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,1,1,0,0,1,0,0)$, $(1,1,1,0,3,3,0,0,0,1,0,1,1)$, $(−14,−14,−14,−12,−1,−1,0,0,−12,1,0,1,1)$, $(−1,0,1,−1,−2,0,0,0,1,0,0,0,0)$, $(0,0,0,14,−12,0,0,0,−14,−1,0,0,1)$, $(1,−1,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,−1,1)$, $(4,0,−4,−1,−2,0,0,0,1,0,0,0,0)$, $(0,0,0,−1,2,0,0,0,1,−1,0,0,1)$, $(−4,−4,−4,2,4,4,0,0,2,1,0,1,1)$, $(0,0,0,0,0,0,−12−3−12,12−3−12,0,0,1,0,0)$, $(0,0,0,0,0,0,12−3−12,−12−3−12,0,0,1,0,0)$
Centralizer type: B^{1}_2+A^{8}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/18 & -1/18\\ -1/18 & 1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$2/5g_{22}+2/5g_{15}+2/5g_{8}-1/5g_{7}+8/5h_{8}+8/5h_{7}-1/5g_{-7}+2/5g_{-8}+2/5g_{-15}+2/5g_{-22}$
matching e: $g_{22}-g_{8}-1/4g_{7}-1/2h_{7}+1/4g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-8/5 & 2/5 & 0 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/5 & -8/5 & 1/5 & 2/5 & 0 & 2/5 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0\\ 0 & 2/5 & -8/5 & 0 & -2/5 & 4/5 & 0 & 0 & 4/5 & 0 & 0 & 0 & 0\\ 2/5 & 2/5 & 0 & 0 & -1/5 & 1/5 & 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0\\ 4/5 & 4/5 & 0 & -2/5 & 0 & 0 & 0 & 0 & 2/5 & 0 & 0 & -4/5 & -4/5\\ 2/5 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & -4/5 & -2/5\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & 2/5 & 0 & 1/5 & -1/5 & 0 & 0 & 0 & 0 & 0 & -2/5 & -2/5\\ 0 & 0 & 0 & -4/5 & 2/5 & -4/5 & 0 & 0 & 0 & 8/5 & 0 & -2/5 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0 & 0 & -2/5 & -1/5 & 0 & 8/5 & -1/5\\ 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & -4/5 & 0 & 0 & -2/5 & 8/5\\ \end{pmatrix}$
$1/2g_{7}-h_{8}-h_{7}+1/2g_{-7}$
matching e: $g_{-8}-g_{-15}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & -1 & 1/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2/5g_{22}+2/5g_{15}+2/5g_{8}-1/5g_{7}+8/5h_{8}+8/5h_{7}-1/5g_{-7}+2/5g_{-8}+2/5g_{-15}+2/5g_{-22}$
matching e: $g_{22}-g_{8}-1/4g_{7}-1/2h_{7}+1/4g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-8/5 & 2/5 & 0 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/5 & -8/5 & 1/5 & 2/5 & 0 & 2/5 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0\\ 0 & 2/5 & -8/5 & 0 & -2/5 & 4/5 & 0 & 0 & 4/5 & 0 & 0 & 0 & 0\\ 2/5 & 2/5 & 0 & 0 & -1/5 & 1/5 & 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0\\ 4/5 & 4/5 & 0 & -2/5 & 0 & 0 & 0 & 0 & 2/5 & 0 & 0 & -4/5 & -4/5\\ 2/5 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & -4/5 & -2/5\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & 2/5 & 0 & 1/5 & -1/5 & 0 & 0 & 0 & 0 & 0 & -2/5 & -2/5\\ 0 & 0 & 0 & -4/5 & 2/5 & -4/5 & 0 & 0 & 0 & 8/5 & 0 & -2/5 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0 & 0 & -2/5 & -1/5 & 0 & 8/5 & -1/5\\ 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & -4/5 & 0 & 0 & -2/5 & 8/5\\ \end{pmatrix}$
$1/2g_{7}-h_{8}-h_{7}+1/2g_{-7}$
matching e: $g_{-8}-g_{-15}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & -1 & 1/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}72 & -36\\ -36 & 36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2/3\sqrt{-3}g_{11}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{3}+2/3\sqrt{-3}g_{-3}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-11}$
matching e: $g_{11}+\left(1/2\sqrt{-3}-1/2\right)g_{4}+\left(-1/2\sqrt{-3}-1/2\right)g_{3}+\left(1/2\sqrt{-3}+1/2\right)g_{-3}+\left(-1/2\sqrt{-3}+1/2\right)g_{-4}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & 0 & -2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2/3\sqrt{-3}g_{11}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{3}+2/3\sqrt{-3}g_{-3}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-11}$
matching e: $g_{11}+\left(1/2\sqrt{-3}-1/2\right)g_{4}+\left(-1/2\sqrt{-3}-1/2\right)g_{3}+\left(1/2\sqrt{-3}+1/2\right)g_{-3}+\left(-1/2\sqrt{-3}+1/2\right)g_{-4}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & 0 & -2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}288\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{37}_1$
h-characteristic: (2, 2, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 74
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $C^{1}_3+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_3+A^{2}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_4+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+3V_{6\omega_{1}}+6V_{5\omega_{1}}+2V_{4\omega_{1}}+4V_{2\omega_{1}}+12V_{\omega_{1}}+22V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&11h_{8}+22h_{7}+22h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&g_{58}+g_{52}+9g_{44}+8g_{17}+5g_{1}\\ f&=&g_{-1}+g_{-17}+g_{-44}+g_{-52}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&11h_{8}+22h_{7}+22h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&2g_{58}+2g_{52}+18g_{44}+16g_{17}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-17}-2g_{-44}-2g_{-52}-2g_{-58}\end{array}$
Centralizer type: $C_3$
Unfold the hidden panel for more information.

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$A^{36}_1$
h-characteristic: (2, 2, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 72
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $C^{1}_3+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_4$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+2V_{6\omega_{1}}+8V_{5\omega_{1}}+V_{4\omega_{1}}+2V_{2\omega_{1}}+8V_{\omega_{1}}+36V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&10h_{8}+20h_{7}+20h_{6}+20h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&g_{58}+9g_{52}+8g_{10}+5g_{1}\\ f&=&g_{-1}+g_{-10}+g_{-52}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&10h_{8}+20h_{7}+20h_{6}+20h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&2g_{58}+18g_{52}+16g_{10}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-10}-2g_{-52}-2g_{-58}\end{array}$
Centralizer type: $C_4$
Unfold the hidden panel for more information.

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$A^{35}_1$
h-characteristic: (2, 2, 1, 0, 0, 0, 0, 0)
Length of the weight dual to h: 70
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $C^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+V_{6\omega_{1}}+10V_{5\omega_{1}}+V_{2\omega_{1}}+55V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&9h_{8}+18h_{7}+18h_{6}+18h_{5}+18h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&9g_{58}+8g_{2}+5g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&9h_{8}+18h_{7}+18h_{6}+18h_{5}+18h_{4}+18h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&18g_{58}+16g_{2}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-58}\end{array}$
Centralizer type: $C_5$
Unfold the hidden panel for more information.

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$A^{32}_1$
h-characteristic: (0, 0, 2, 0, 0, 0, 0, 2)
Length of the weight dual to h: 64
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 7
Containing regular semisimple subalgebra number 1: $3B^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_3+B^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $3B^{1}_2+A^{2}_1$ Containing regular semisimple subalgebra number 4: $A^{2}_3+B^{1}_2+A^{2}_1$ Containing regular semisimple subalgebra number 5: $2C^{1}_3+B^{1}_2$ Containing regular semisimple subalgebra number 6: $C^{1}_3+2B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 7: $A^{2}_3+C^{1}_3+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $6V_{6\omega_{1}}+9V_{4\omega_{1}}+15V_{2\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&14h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{52}+4g_{44}+4g_{34}+g_{22}+3g_{18}+3g_{17}+3g_{16}+g_{8}\\ f&=&g_{-8}+g_{-16}+g_{-17}+g_{-18}+g_{-22}+g_{-34}+g_{-44}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&14h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{52}+8g_{44}+8g_{34}+2g_{22}+6g_{18}+6g_{17}+6g_{16}+2g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-16}-2g_{-17}-2g_{-18}-2g_{-22}-2g_{-34}-2g_{-44}-2g_{-52}\end{array}$
Centralizer type: $A^{16}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 4): $g_{4}+g_{1}-g_{-1}-g_{-4}$, $g_{5}+g_{2}-g_{-2}-g_{-5}$, $g_{7}-g_{-7}$, $g_{12}+g_{9}-g_{-9}-g_{-12}$
Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 2): $g_{12}+g_{9}+g_{5}+g_{4}+g_{2}+g_{1}-g_{-1}-g_{-2}-g_{-4}-g_{-5}-g_{-9}-g_{-12}$, $g_{7}-g_{-7}$
Cartan-generating semisimple element: $g_{12}+g_{9}+g_{7}+g_{5}+g_{4}+g_{2}+g_{1}-g_{-1}-g_{-2}-g_{-4}-g_{-5}-g_{-7}-g_{-9}-g_{-12}$
adjoint action: $\begin{pmatrix}0 & -1 & 0 & 1\\ 1 & 0 & 0 & -1\\ 0 & 0 & 0 & 0\\ -1 & 1 & 0 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^4+3x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x^2+3)
Eigenvalues of ad H: $0$, $\sqrt{-3}$, $-\sqrt{-3}$
4 eigenvectors of ad H: $(0,0,1,0)$, $(1,1,0,1)$, $(−12−3−12,12−3−12,0,1)$, $(12−3−12,−12−3−12,0,1)$
Centralizer type: A^{16}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/144\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2/3\sqrt{-3}g_{12}-2/3\sqrt{-3}g_{9}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{2}-2/3\sqrt{-3}g_{1}+2/3\sqrt{-3}g_{-1}+2/3\sqrt{-3}g_{-2}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-9}+2/3\sqrt{-3}g_{-12}$
matching e: $g_{12}+g_{9}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}-1/2\right)g_{2}+\left(-1/2\sqrt{-3}-1/2\right)g_{1}+\left(1/2\sqrt{-3}+1/2\right)g_{-1}+\left(-1/2\sqrt{-3}+1/2\right)g_{-2}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-9}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0\\ 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2/3\sqrt{-3}g_{12}-2/3\sqrt{-3}g_{9}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{2}-2/3\sqrt{-3}g_{1}+2/3\sqrt{-3}g_{-1}+2/3\sqrt{-3}g_{-2}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-9}+2/3\sqrt{-3}g_{-12}$
matching e: $g_{12}+g_{9}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}-1/2\right)g_{2}+\left(-1/2\sqrt{-3}-1/2\right)g_{1}+\left(1/2\sqrt{-3}+1/2\right)g_{-1}+\left(-1/2\sqrt{-3}+1/2\right)g_{-2}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-9}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0\\ 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}576\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{31}_1$
h-characteristic: (0, 0, 2, 0, 0, 0, 1, 0)
Length of the weight dual to h: 62
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $3B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_3+B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_3+2B^{1}_2$ Containing regular semisimple subalgebra number 4: $A^{2}_3+C^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $6V_{6\omega_{1}}+6V_{4\omega_{1}}+6V_{3\omega_{1}}+10V_{2\omega_{1}}+2V_{\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&13h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{52}+4g_{44}+4g_{34}+g_{22}+3g_{18}+3g_{17}+3g_{16}\\ f&=&g_{-16}+g_{-17}+g_{-18}+g_{-22}+g_{-34}+g_{-44}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&13h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{52}+8g_{44}+8g_{34}+2g_{22}+6g_{18}+6g_{17}+6g_{16}\\ [h, f]&=&-2g_{-16}-2g_{-17}-2g_{-18}-2g_{-22}-2g_{-34}-2g_{-44}-2g_{-52}\end{array}$
Centralizer type: $A^{16}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 6): $g_{-8}$, $h_{8}$, $g_{4}+g_{1}-g_{-1}-g_{-4}$, $g_{5}+g_{2}-g_{-2}-g_{-5}$, $g_{8}$, $g_{12}+g_{9}-g_{-9}-g_{-12}$
Basis of centralizer intersected with cartan (dimension: 1): $-h_{8}$
Cartan of centralizer (dimension: 2): $-g_{8}-h_{8}-g_{-8}$, $g_{12}+g_{9}+g_{5}+g_{4}+g_{2}+g_{1}-g_{-1}-g_{-2}-g_{-4}-g_{-5}-g_{-9}-g_{-12}$
Cartan-generating semisimple element: $g_{12}+g_{9}+g_{8}+g_{5}+g_{4}+g_{2}+g_{1}+h_{8}-g_{-1}-g_{-2}-g_{-4}-g_{-5}+g_{-8}-g_{-9}-g_{-12}$
adjoint action: $\begin{pmatrix}-2 & 2 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & -1\\ 0 & -2 & 0 & 0 & 2 & 0\\ 0 & 0 & -1 & 1 & 0 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6-5x^4-24x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x^2-8)(x^2+3)
Eigenvalues of ad H: $0$, $2\sqrt{2}$, $-2\sqrt{2}$, $\sqrt{-3}$, $-\sqrt{-3}$
6 eigenvectors of ad H: $(1,1,0,0,1,0)$, $(0,0,1,1,0,1)$, $(22−3,−2+1,0,0,1,0)$, $(−22−3,2+1,0,0,1,0)$, $(0,0,−12−3−12,12−3−12,0,1)$, $(0,0,12−3−12,−12−3−12,0,1)$
Centralizer type: A^{16}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$1/2\sqrt{2}g_{8}+1/2\sqrt{2}h_{8}+1/2\sqrt{2}g_{-8}$
matching e: $g_{8}+\left(-\sqrt{2}+1\right)h_{8}+\left(2\sqrt{2}-3\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & -1/2\sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$1/2\sqrt{2}g_{8}+1/2\sqrt{2}h_{8}+1/2\sqrt{2}g_{-8}$
matching e: $g_{8}+\left(-\sqrt{2}+1\right)h_{8}+\left(2\sqrt{2}-3\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & -1/2\sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/144\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2/3\sqrt{-3}g_{12}-2/3\sqrt{-3}g_{9}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{2}-2/3\sqrt{-3}g_{1}+2/3\sqrt{-3}g_{-1}+2/3\sqrt{-3}g_{-2}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-9}+2/3\sqrt{-3}g_{-12}$
matching e: $g_{12}+g_{9}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}-1/2\right)g_{2}+\left(-1/2\sqrt{-3}-1/2\right)g_{1}+\left(1/2\sqrt{-3}+1/2\right)g_{-1}+\left(-1/2\sqrt{-3}+1/2\right)g_{-2}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-9}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2/3\sqrt{-3}g_{12}-2/3\sqrt{-3}g_{9}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{2}-2/3\sqrt{-3}g_{1}+2/3\sqrt{-3}g_{-1}+2/3\sqrt{-3}g_{-2}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-9}+2/3\sqrt{-3}g_{-12}$
matching e: $g_{12}+g_{9}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}-1/2\right)g_{2}+\left(-1/2\sqrt{-3}-1/2\right)g_{1}+\left(1/2\sqrt{-3}+1/2\right)g_{-1}+\left(-1/2\sqrt{-3}+1/2\right)g_{-2}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-9}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}576\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{30}_1$
h-characteristic: (0, 0, 2, 0, 0, 1, 0, 0)
Length of the weight dual to h: 60
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $3B^{1}_2$ Containing regular semisimple subalgebra number 2: $A^{2}_3+B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $6V_{6\omega_{1}}+3V_{4\omega_{1}}+12V_{3\omega_{1}}+6V_{2\omega_{1}}+13V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+24h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{52}+4g_{44}+4g_{34}+3g_{18}+3g_{17}+3g_{16}\\ f&=&g_{-16}+g_{-17}+g_{-18}+g_{-34}+g_{-44}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+24h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{52}+8g_{44}+8g_{34}+6g_{18}+6g_{17}+6g_{16}\\ [h, f]&=&-2g_{-16}-2g_{-17}-2g_{-18}-2g_{-34}-2g_{-44}-2g_{-52}\end{array}$
Centralizer type: $B_2+A^{16}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 13): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $h_{7}$, $h_{8}$, $g_{4}+g_{1}-g_{-1}-g_{-4}$, $g_{5}+g_{2}-g_{-2}-g_{-5}$, $g_{7}$, $g_{8}$, $g_{12}+g_{9}-g_{-9}-g_{-12}$, $g_{15}$, $g_{22}$
Basis of centralizer intersected with cartan (dimension: 2): $-h_{8}$, $-h_{7}$
Cartan of centralizer (dimension: 3): $-g_{22}-g_{15}-g_{8}-3h_{8}-3h_{7}-g_{-8}-g_{-15}-g_{-22}$, $-g_{7}+2h_{8}+2h_{7}-g_{-7}$, $g_{12}+g_{9}+g_{5}+g_{4}+g_{2}+g_{1}-g_{-1}-g_{-2}-g_{-4}-g_{-5}-g_{-9}-g_{-12}$
Cartan-generating semisimple element: $g_{22}+g_{15}+g_{12}+g_{9}+g_{8}+g_{7}+g_{5}+g_{4}+g_{2}+g_{1}+h_{8}+h_{7}-g_{-1}-g_{-2}-g_{-4}-g_{-5}+g_{-7}+g_{-8}-g_{-9}-g_{-12}+g_{-15}+g_{-22}$
adjoint action: $\begin{pmatrix}-1 & -2 & 0 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & -1 & -1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & -2 & -1 & 0 & -1 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & -1 & 0 & -1 & 0\\ 2 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & -2 & -2\\ 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -2 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 1 & 1 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & -1\\ 0 & 0 & 0 & -2 & 1 & -2 & 0 & 0 & 0 & 1 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & -1 & 0 & 0 & -1 & 1 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -2 & 0 & 0 & 2 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{13}-36x^{11}+282x^9-64x^7-2883x^5+2700x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -5)(x -3)(x -2)(x -1)(x +1)(x +2)(x +3)(x +5)(x^2+3)
Eigenvalues of ad H: $0$, $5$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-5$, $\sqrt{-3}$, $-\sqrt{-3}$
13 eigenvectors of ad H: $(0,0,0,1,−2,−2,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,1,1,0,0,1,0,0)$, $(1,1,1,0,3,3,0,0,0,1,0,1,1)$, $(−14,−14,−14,−12,−1,−1,0,0,−12,1,0,1,1)$, $(−1,0,1,−1,−2,0,0,0,1,0,0,0,0)$, $(0,0,0,14,−12,0,0,0,−14,−1,0,0,1)$, $(1,−1,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,−1,1)$, $(4,0,−4,−1,−2,0,0,0,1,0,0,0,0)$, $(0,0,0,−1,2,0,0,0,1,−1,0,0,1)$, $(−4,−4,−4,2,4,4,0,0,2,1,0,1,1)$, $(0,0,0,0,0,0,−12−3−12,12−3−12,0,0,1,0,0)$, $(0,0,0,0,0,0,12−3−12,−12−3−12,0,0,1,0,0)$
Centralizer type: B^{1}_2+A^{16}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/18 & -1/18\\ -1/18 & 1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$2/5g_{22}+2/5g_{15}+2/5g_{8}-1/5g_{7}+8/5h_{8}+8/5h_{7}-1/5g_{-7}+2/5g_{-8}+2/5g_{-15}+2/5g_{-22}$
matching e: $g_{22}-g_{8}-1/4g_{7}-1/2h_{7}+1/4g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-8/5 & 2/5 & 0 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/5 & -8/5 & 1/5 & 2/5 & 0 & 2/5 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0\\ 0 & 2/5 & -8/5 & 0 & -2/5 & 4/5 & 0 & 0 & 4/5 & 0 & 0 & 0 & 0\\ 2/5 & 2/5 & 0 & 0 & -1/5 & 1/5 & 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0\\ 4/5 & 4/5 & 0 & -2/5 & 0 & 0 & 0 & 0 & 2/5 & 0 & 0 & -4/5 & -4/5\\ 2/5 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & -4/5 & -2/5\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & 2/5 & 0 & 1/5 & -1/5 & 0 & 0 & 0 & 0 & 0 & -2/5 & -2/5\\ 0 & 0 & 0 & -4/5 & 2/5 & -4/5 & 0 & 0 & 0 & 8/5 & 0 & -2/5 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0 & 0 & -2/5 & -1/5 & 0 & 8/5 & -1/5\\ 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & -4/5 & 0 & 0 & -2/5 & 8/5\\ \end{pmatrix}$
$1/2g_{7}-h_{8}-h_{7}+1/2g_{-7}$
matching e: $g_{-8}-g_{-15}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & -1 & 1/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2/5g_{22}+2/5g_{15}+2/5g_{8}-1/5g_{7}+8/5h_{8}+8/5h_{7}-1/5g_{-7}+2/5g_{-8}+2/5g_{-15}+2/5g_{-22}$
matching e: $g_{22}-g_{8}-1/4g_{7}-1/2h_{7}+1/4g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-8/5 & 2/5 & 0 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/5 & -8/5 & 1/5 & 2/5 & 0 & 2/5 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0\\ 0 & 2/5 & -8/5 & 0 & -2/5 & 4/5 & 0 & 0 & 4/5 & 0 & 0 & 0 & 0\\ 2/5 & 2/5 & 0 & 0 & -1/5 & 1/5 & 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0\\ 4/5 & 4/5 & 0 & -2/5 & 0 & 0 & 0 & 0 & 2/5 & 0 & 0 & -4/5 & -4/5\\ 2/5 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & -4/5 & -2/5\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & 2/5 & 0 & 1/5 & -1/5 & 0 & 0 & 0 & 0 & 0 & -2/5 & -2/5\\ 0 & 0 & 0 & -4/5 & 2/5 & -4/5 & 0 & 0 & 0 & 8/5 & 0 & -2/5 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0 & 0 & -2/5 & -1/5 & 0 & 8/5 & -1/5\\ 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & -4/5 & 0 & 0 & -2/5 & 8/5\\ \end{pmatrix}$
$1/2g_{7}-h_{8}-h_{7}+1/2g_{-7}$
matching e: $g_{-8}-g_{-15}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & -1 & 1/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}72 & -36\\ -36 & 36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/144\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2/3\sqrt{-3}g_{12}-2/3\sqrt{-3}g_{9}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{2}-2/3\sqrt{-3}g_{1}+2/3\sqrt{-3}g_{-1}+2/3\sqrt{-3}g_{-2}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-9}+2/3\sqrt{-3}g_{-12}$
matching e: $g_{12}+g_{9}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}-1/2\right)g_{2}+\left(-1/2\sqrt{-3}-1/2\right)g_{1}+\left(1/2\sqrt{-3}+1/2\right)g_{-1}+\left(-1/2\sqrt{-3}+1/2\right)g_{-2}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-9}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & 0 & -2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2/3\sqrt{-3}g_{12}-2/3\sqrt{-3}g_{9}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{2}-2/3\sqrt{-3}g_{1}+2/3\sqrt{-3}g_{-1}+2/3\sqrt{-3}g_{-2}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-9}+2/3\sqrt{-3}g_{-12}$
matching e: $g_{12}+g_{9}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}-1/2\right)g_{2}+\left(-1/2\sqrt{-3}-1/2\right)g_{1}+\left(1/2\sqrt{-3}+1/2\right)g_{-1}+\left(-1/2\sqrt{-3}+1/2\right)g_{-2}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-9}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & 0 & -2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}576\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{29}_1$
h-characteristic: (0, 1, 0, 1, 0, 0, 1, 0)
Length of the weight dual to h: 58
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $A^{2}_2+2B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_3+A^{2}_2+A^{1}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_3+A^{2}_2+B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{6\omega_{1}}+4V_{5\omega_{1}}+6V_{4\omega_{1}}+6V_{3\omega_{1}}+7V_{2\omega_{1}}+6V_{\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&13h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{44}+2g_{37}+4g_{34}+2g_{26}+3g_{24}+3g_{23}+g_{22}\\ f&=&g_{-22}+g_{-23}+g_{-24}+g_{-26}+g_{-34}+g_{-37}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&13h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{44}+4g_{37}+8g_{34}+4g_{26}+6g_{24}+6g_{23}+2g_{22}\\ [h, f]&=&-2g_{-22}-2g_{-23}-2g_{-24}-2g_{-26}-2g_{-34}-2g_{-37}-2g_{-44}\end{array}$
Centralizer type: $A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 4): $h_{8}-2h_{3}$, $g_{3}+g_{-8}$, $g_{5}+g_{1}-g_{-1}-g_{-5}$, $g_{8}+g_{-3}$
Basis of centralizer intersected with cartan (dimension: 1): $1/2h_{8}-h_{3}$
Cartan of centralizer (dimension: 2): $1/2h_{8}-h_{3}$, $g_{5}+g_{1}-g_{-1}-g_{-5}$
Cartan-generating semisimple element: $1/2h_{8}-h_{3}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^4-x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -1)(x +1)
Eigenvalues of ad H: $0$, $1$, $-1$
4 eigenvectors of ad H: $(1,0,0,0)$, $(0,0,1,0)$, $(0,0,0,1)$, $(0,1,0,0)$
Centralizer type: A^{3}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}-2h_{3}$
matching e: $g_{8}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}-2h_{3}$
matching e: $g_{8}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{28}_1$
h-characteristic: (0, 1, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 56
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{2}_2+2B^{1}_2$ Containing regular semisimple subalgebra number 2: $A^{2}_3+A^{2}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{6\omega_{1}}+4V_{5\omega_{1}}+4V_{4\omega_{1}}+8V_{3\omega_{1}}+8V_{2\omega_{1}}+4V_{\omega_{1}}+7V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+24h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{44}+2g_{42}+4g_{34}+3g_{24}+3g_{23}+2g_{19}\\ f&=&g_{-19}+g_{-23}+g_{-24}+g_{-34}+g_{-42}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+24h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{44}+4g_{42}+8g_{34}+6g_{24}+6g_{23}+4g_{19}\\ [h, f]&=&-2g_{-19}-2g_{-23}-2g_{-24}-2g_{-34}-2g_{-42}-2g_{-44}\end{array}$
Centralizer type: $A^{3}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 7): $g_{-8}$, $h_{7}-h_{3}$, $h_{8}$, $g_{3}+g_{-22}$, $g_{5}+g_{1}-g_{-1}-g_{-5}$, $g_{8}$, $g_{22}+g_{-3}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{7}-h_{3}$, $-h_{8}$
Cartan of centralizer (dimension: 3): $g_{5}+g_{1}-g_{-1}-g_{-5}$, $-h_{8}$, $h_{7}-h_{3}$
Cartan-generating semisimple element: $-h_{8}+h_{7}-h_{3}$
adjoint action: $\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^7-10x^5+9x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -3)(x -1)(x +1)(x +3)
Eigenvalues of ad H: $0$, $3$, $1$, $-1$, $-3$
7 eigenvectors of ad H: $(0,1,0,0,0,0,0)$, $(0,0,1,0,0,0,0)$, $(0,0,0,0,1,0,0)$, $(1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1)$, $(0,0,0,1,0,0,0)$, $(0,0,0,0,0,1,0)$
Centralizer type: A^{3}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}-2h_{3}$
matching e: $g_{22}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}-2h_{3}$
matching e: $g_{22}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{26}_1$
h-characteristic: (1, 0, 0, 0, 1, 1, 0, 0)
Length of the weight dual to h: 52
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $2A^{2}_2+B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+4V_{5\omega_{1}}+10V_{4\omega_{1}}+4V_{3\omega_{1}}+7V_{2\omega_{1}}+4V_{\omega_{1}}+10V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+24h_{7}+24h_{6}+22h_{5}+18h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&2g_{46}+4g_{34}+2g_{32}+3g_{29}+2g_{25}+2g_{20}\\ f&=&g_{-20}+g_{-25}+g_{-29}+g_{-32}+g_{-34}+g_{-46}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+24h_{7}+24h_{6}+22h_{5}+18h_{4}+14h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&4g_{46}+8g_{34}+4g_{32}+6g_{29}+4g_{25}+4g_{20}\\ [h, f]&=&-2g_{-20}-2g_{-25}-2g_{-29}-2g_{-32}-2g_{-34}-2g_{-46}\end{array}$
Centralizer type: $B^{3}_2$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 10): $h_{7}+h_{4}-h_{2}$, $h_{8}-2h_{4}$, $g_{2}+g_{-22}$, $g_{4}+g_{-8}$, $g_{8}+g_{-4}$, $g_{10}-g_{-7}-g_{-11}$, $g_{11}+g_{7}-g_{-10}$, $g_{15}+g_{3}+g_{-17}$, $g_{17}+g_{-3}+g_{-15}$, $g_{22}+g_{-2}$
Basis of centralizer intersected with cartan (dimension: 2): $1/2h_{8}-h_{4}$, $1/2h_{8}+h_{7}-h_{2}$
Cartan of centralizer (dimension: 2): $1/2h_{8}-h_{4}$, $1/2h_{8}+h_{7}-h_{2}$
Cartan-generating semisimple element: $5h_{8}+9h_{7}-h_{4}-9h_{2}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{10}-123x^8+3843x^6-36121x^4+32400x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -9)(x -5)(x -4)(x -1)(x +1)(x +4)(x +5)(x +9)
Eigenvalues of ad H: $0$, $9$, $5$, $4$, $1$, $-1$, $-4$, $-5$, $-9$
10 eigenvectors of ad H: $(1,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0)$, $(0,0,1,0,0,0,0,0,0,0)$
Centralizer type: B^{3}_2
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/54 & -1/54\\ -1/54 & 1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$2h_{7}+2h_{4}-2h_{2}$
matching e: $g_{11}+g_{7}-g_{-10}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$h_{8}-2h_{4}$
matching e: $g_{8}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{7}+2h_{4}-2h_{2}$
matching e: $g_{11}+g_{7}-g_{-10}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$h_{8}-2h_{4}$
matching e: $g_{8}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}216 & -108\\ -108 & 108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{24}_1$
h-characteristic: (0, 2, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 48
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 10
Containing regular semisimple subalgebra number 1: $2B^{1}_2+4A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_3+4A^{1}_1$ Containing regular semisimple subalgebra number 3: $2B^{1}_2+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $A^{2}_3+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 5: $2B^{1}_2+2A^{2}_1$ Containing regular semisimple subalgebra number 6: $A^{2}_3+2A^{2}_1$ Containing regular semisimple subalgebra number 7: $2C^{1}_3+2A^{1}_1$ Containing regular semisimple subalgebra number 8: $C^{1}_3+B^{1}_2+3A^{1}_1$ Containing regular semisimple subalgebra number 9: $2C^{1}_3+A^{2}_1$ Containing regular semisimple subalgebra number 10: $C^{1}_3+B^{1}_2+A^{2}_1+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{6\omega_{1}}+9V_{4\omega_{1}}+21V_{2\omega_{1}}+7V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+22h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{58}+4g_{52}+g_{44}+g_{34}+g_{22}+3g_{10}+3g_{9}+g_{8}\\ f&=&g_{-8}+g_{-9}+g_{-10}+g_{-22}+g_{-34}+g_{-44}+g_{-52}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+22h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{58}+8g_{52}+2g_{44}+2g_{34}+2g_{22}+6g_{10}+6g_{9}+2g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-9}-2g_{-10}-2g_{-22}-2g_{-34}-2g_{-44}-2g_{-52}-2g_{-58}\end{array}$
Centralizer type: $2A^{4}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 7): $g_{3}+g_{1}-g_{-1}-g_{-3}$, $g_{5}-g_{-5}$, $g_{6}-g_{-6}$, $g_{7}-g_{-7}$, $g_{13}-g_{-13}$, $g_{14}-g_{-14}$, $g_{20}-g_{-20}$
Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 3): $g_{3}+g_{1}-g_{-1}-g_{-3}$, $g_{20}+g_{7}+g_{6}+g_{5}-g_{-5}-g_{-6}-g_{-7}-g_{-20}$, $g_{14}+g_{13}-g_{-13}-g_{-14}$
Cartan-generating semisimple element: $g_{20}+g_{14}+g_{13}+g_{7}+g_{6}+g_{5}+g_{3}+g_{1}-g_{-1}-g_{-3}-g_{-5}-g_{-6}-g_{-7}-g_{-13}-g_{-14}-g_{-20}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 1 & -1 & 1\\ 0 & 1 & 0 & -1 & -1 & 1 & 0\\ 0 & 0 & 1 & 0 & 1 & -1 & -1\\ 0 & -1 & 1 & -1 & 0 & 0 & 1\\ 0 & 1 & -1 & 1 & 0 & 0 & -1\\ 0 & -1 & 0 & 1 & -1 & 1 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^7+12x^5+32x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x^2+4)(x^2+8)
Eigenvalues of ad H: $0$, $2\sqrt{-1}$, $-2\sqrt{-1}$, $2\sqrt{-2}$, $-2\sqrt{-2}$
7 eigenvectors of ad H: $(1,0,0,0,0,0,0)$, $(0,0,0,0,1,1,0)$, $(0,1,1,1,0,0,1)$, $(0,−−1,−1,−1,0,0,1)$, $(0,−1,−1,−−1,0,0,1)$, $(0,−1,1,−1,−−2,−2,1)$, $(0,−1,1,−1,−2,−−2,1)$
Centralizer type: 2A^{4}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-\sqrt{-1}g_{14}-\sqrt{-1}g_{13}+\sqrt{-1}g_{-13}+\sqrt{-1}g_{-14}$
matching e: $g_{20}+\sqrt{-1}g_{7}-g_{6}-\sqrt{-1}g_{5}+\sqrt{-1}g_{-5}+g_{-6}-\sqrt{-1}g_{-7}-g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & -\sqrt{-1}\\ 0 & -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0\\ 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-\sqrt{-1}g_{14}-\sqrt{-1}g_{13}+\sqrt{-1}g_{-13}+\sqrt{-1}g_{-14}$
matching e: $g_{20}+\sqrt{-1}g_{7}-g_{6}-\sqrt{-1}g_{5}+\sqrt{-1}g_{-5}+g_{-6}-\sqrt{-1}g_{-7}-g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & -\sqrt{-1}\\ 0 & -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0\\ 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-1/2\sqrt{-2}g_{20}-1/2\sqrt{-2}g_{7}-1/2\sqrt{-2}g_{6}-1/2\sqrt{-2}g_{5}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-6}+1/2\sqrt{-2}g_{-7}+1/2\sqrt{-2}g_{-20}$
matching e: $g_{20}+\sqrt{-2}g_{14}-\sqrt{-2}g_{13}-g_{7}+g_{6}-g_{5}+g_{-5}-g_{-6}+g_{-7}+\sqrt{-2}g_{-13}-\sqrt{-2}g_{-14}-g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & -1/2\sqrt{-2}\\ 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 1/2\sqrt{-2}\\ 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-1/2\sqrt{-2}g_{20}-1/2\sqrt{-2}g_{7}-1/2\sqrt{-2}g_{6}-1/2\sqrt{-2}g_{5}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-6}+1/2\sqrt{-2}g_{-7}+1/2\sqrt{-2}g_{-20}$
matching e: $g_{20}+\sqrt{-2}g_{14}-\sqrt{-2}g_{13}-g_{7}+g_{6}-g_{5}+g_{-5}-g_{-6}+g_{-7}+\sqrt{-2}g_{-13}-\sqrt{-2}g_{-14}-g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & -1/2\sqrt{-2}\\ 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 1/2\sqrt{-2}\\ 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{23}_1$
h-characteristic: (0, 2, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 46
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 7
Containing regular semisimple subalgebra number 1: $2B^{1}_2+3A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_3+3A^{1}_1$ Containing regular semisimple subalgebra number 3: $2B^{1}_2+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 4: $A^{2}_3+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 5: $2C^{1}_3+A^{1}_1$ Containing regular semisimple subalgebra number 6: $C^{1}_3+B^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 7: $C^{1}_3+B^{1}_2+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{6\omega_{1}}+7V_{4\omega_{1}}+4V_{3\omega_{1}}+15V_{2\omega_{1}}+6V_{\omega_{1}}+7V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&11h_{8}+22h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{58}+4g_{52}+g_{44}+g_{34}+g_{22}+3g_{10}+3g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-22}+g_{-34}+g_{-44}+g_{-52}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&11h_{8}+22h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{58}+8g_{52}+2g_{44}+2g_{34}+2g_{22}+6g_{10}+6g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-22}-2g_{-34}-2g_{-44}-2g_{-52}-2g_{-58}\end{array}$
Centralizer type: $A^{8}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 7): $g_{-8}$, $h_{8}$, $g_{3}+g_{1}-g_{-1}-g_{-3}$, $g_{5}-g_{-5}$, $g_{6}-g_{-6}$, $g_{8}$, $g_{13}-g_{-13}$
Basis of centralizer intersected with cartan (dimension: 1): $-h_{8}$
Cartan of centralizer (dimension: 3): $-g_{8}-h_{8}-g_{-8}$, $g_{13}+g_{6}+g_{5}-g_{-5}-g_{-6}-g_{-13}$, $g_{3}+g_{1}-g_{-1}-g_{-3}$
Cartan-generating semisimple element: $g_{13}+g_{8}+g_{6}+g_{5}+g_{3}+g_{1}+h_{8}-g_{-1}-g_{-3}-g_{-5}-g_{-6}+g_{-8}-g_{-13}$
adjoint action: $\begin{pmatrix}-2 & 2 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 1\\ 0 & 0 & 0 & 1 & 0 & 0 & -1\\ 0 & -2 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & -1 & 1 & 0 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^7-5x^5-24x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x^2-8)(x^2+3)
Eigenvalues of ad H: $0$, $2\sqrt{2}$, $-2\sqrt{2}$, $\sqrt{-3}$, $-\sqrt{-3}$
7 eigenvectors of ad H: $(0,0,1,0,0,0,0)$, $(1,1,0,0,0,1,0)$, $(0,0,0,1,1,0,1)$, $(22−3,−2+1,0,0,0,1,0)$, $(−22−3,2+1,0,0,0,1,0)$, $(0,0,0,−12−3−12,12−3−12,0,1)$, $(0,0,0,12−3−12,−12−3−12,0,1)$
Centralizer type: A^{8}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$1/2\sqrt{2}g_{8}+1/2\sqrt{2}h_{8}+1/2\sqrt{2}g_{-8}$
matching e: $g_{8}+\left(-\sqrt{2}+1\right)h_{8}+\left(2\sqrt{2}-3\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & 0 & -1/2\sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$1/2\sqrt{2}g_{8}+1/2\sqrt{2}h_{8}+1/2\sqrt{2}g_{-8}$
matching e: $g_{8}+\left(-\sqrt{2}+1\right)h_{8}+\left(2\sqrt{2}-3\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & 0 & -1/2\sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2/3\sqrt{-3}g_{13}-2/3\sqrt{-3}g_{6}-2/3\sqrt{-3}g_{5}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-6}+2/3\sqrt{-3}g_{-13}$
matching e: $g_{13}+\left(1/2\sqrt{-3}-1/2\right)g_{6}+\left(-1/2\sqrt{-3}-1/2\right)g_{5}+\left(1/2\sqrt{-3}+1/2\right)g_{-5}+\left(-1/2\sqrt{-3}+1/2\right)g_{-6}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2/3\sqrt{-3}g_{13}-2/3\sqrt{-3}g_{6}-2/3\sqrt{-3}g_{5}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-6}+2/3\sqrt{-3}g_{-13}$
matching e: $g_{13}+\left(1/2\sqrt{-3}-1/2\right)g_{6}+\left(-1/2\sqrt{-3}-1/2\right)g_{5}+\left(1/2\sqrt{-3}+1/2\right)g_{-5}+\left(-1/2\sqrt{-3}+1/2\right)g_{-6}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}288\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{22}_1$
h-characteristic: (0, 2, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 44
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 6
Containing regular semisimple subalgebra number 1: $2B^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_3+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $2B^{1}_2+A^{2}_1$ Containing regular semisimple subalgebra number 4: $A^{2}_3+A^{2}_1$ Containing regular semisimple subalgebra number 5: $2C^{1}_3$ Containing regular semisimple subalgebra number 6: $C^{1}_3+B^{1}_2+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{6\omega_{1}}+5V_{4\omega_{1}}+8V_{3\omega_{1}}+10V_{2\omega_{1}}+8V_{\omega_{1}}+12V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&10h_{8}+20h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{58}+4g_{52}+g_{44}+g_{34}+3g_{10}+3g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-34}+g_{-44}+g_{-52}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&10h_{8}+20h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{58}+8g_{52}+2g_{44}+2g_{34}+6g_{10}+6g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-34}-2g_{-44}-2g_{-52}-2g_{-58}\end{array}$
Centralizer type: $B_2$
Unfold the hidden panel for more information.

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$A^{21}_1$
h-characteristic: (1, 0, 1, 0, 0, 0, 1, 0)
Length of the weight dual to h: 42
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $A^{2}_2+B^{1}_2+3A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_2+B^{1}_2+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_3+A^{2}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_3+A^{2}_2+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+2V_{5\omega_{1}}+6V_{4\omega_{1}}+8V_{3\omega_{1}}+11V_{2\omega_{1}}+8V_{\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&11h_{8}+22h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&4g_{52}+g_{44}+2g_{41}+g_{34}+2g_{31}+g_{22}+3g_{16}\\ f&=&g_{-16}+g_{-22}+g_{-31}+g_{-34}+g_{-41}+g_{-44}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&11h_{8}+22h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&8g_{52}+2g_{44}+4g_{41}+2g_{34}+4g_{31}+2g_{22}+6g_{16}\\ [h, f]&=&-2g_{-16}-2g_{-22}-2g_{-31}-2g_{-34}-2g_{-41}-2g_{-44}-2g_{-52}\end{array}$
Centralizer type: $A^{8}_1+A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 6): $h_{8}-2h_{2}$, $g_{2}+g_{-8}$, $g_{5}-g_{-5}$, $g_{6}-g_{-6}$, $g_{8}+g_{-2}$, $g_{13}-g_{-13}$
Basis of centralizer intersected with cartan (dimension: 1): $1/2h_{8}-h_{2}$
Cartan of centralizer (dimension: 2): $1/2g_{8}+1/2g_{2}+1/2h_{8}-h_{2}+1/2g_{-2}+1/2g_{-8}$, $g_{13}+g_{6}+g_{5}-g_{-5}-g_{-6}-g_{-13}$
Cartan-generating semisimple element: $g_{13}+g_{8}+g_{6}+g_{5}+g_{2}+h_{8}-2h_{2}+g_{-2}-g_{-5}-g_{-6}+g_{-8}-g_{-13}$
adjoint action: $\begin{pmatrix}0 & 1 & 0 & 0 & -1 & 0\\ 2 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & -1\\ -2 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & -1 & 1 & 0 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6-5x^4-24x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x^2-8)(x^2+3)
Eigenvalues of ad H: $0$, $2\sqrt{2}$, $-2\sqrt{2}$, $\sqrt{-3}$, $-\sqrt{-3}$
6 eigenvectors of ad H: $(1,1,0,0,1,0)$, $(0,0,1,1,0,1)$, $(−2+1,22−3,0,0,1,0)$, $(2+1,−22−3,0,0,1,0)$, $(0,0,−12−3−12,12−3−12,0,1)$, $(0,0,12−3−12,−12−3−12,0,1)$
Centralizer type: A^{8}_1+A^{3}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$1/2\sqrt{2}g_{8}+1/2\sqrt{2}g_{2}+1/2\sqrt{2}h_{8}-\sqrt{2}h_{2}+1/2\sqrt{2}g_{-2}+1/2\sqrt{2}g_{-8}$
matching e: $g_{8}+\left(2\sqrt{2}-3\right)g_{2}+\left(-\sqrt{2}+1\right)h_{8}+\left(2\sqrt{2}-2\right)h_{2}+g_{-2}+\left(2\sqrt{2}-3\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 1/2\sqrt{2} & 0 & 0 & -1/2\sqrt{2} & 0\\ \sqrt{2} & -\sqrt{2} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ -\sqrt{2} & 0 & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$1/2\sqrt{2}g_{8}+1/2\sqrt{2}g_{2}+1/2\sqrt{2}h_{8}-\sqrt{2}h_{2}+1/2\sqrt{2}g_{-2}+1/2\sqrt{2}g_{-8}$
matching e: $g_{8}+\left(2\sqrt{2}-3\right)g_{2}+\left(-\sqrt{2}+1\right)h_{8}+\left(2\sqrt{2}-2\right)h_{2}+g_{-2}+\left(2\sqrt{2}-3\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 1/2\sqrt{2} & 0 & 0 & -1/2\sqrt{2} & 0\\ \sqrt{2} & -\sqrt{2} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ -\sqrt{2} & 0 & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2/3\sqrt{-3}g_{13}-2/3\sqrt{-3}g_{6}-2/3\sqrt{-3}g_{5}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-6}+2/3\sqrt{-3}g_{-13}$
matching e: $g_{13}+\left(1/2\sqrt{-3}-1/2\right)g_{6}+\left(-1/2\sqrt{-3}-1/2\right)g_{5}+\left(1/2\sqrt{-3}+1/2\right)g_{-5}+\left(-1/2\sqrt{-3}+1/2\right)g_{-6}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2/3\sqrt{-3}g_{13}-2/3\sqrt{-3}g_{6}-2/3\sqrt{-3}g_{5}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-6}+2/3\sqrt{-3}g_{-13}$
matching e: $g_{13}+\left(1/2\sqrt{-3}-1/2\right)g_{6}+\left(-1/2\sqrt{-3}-1/2\right)g_{5}+\left(1/2\sqrt{-3}+1/2\right)g_{-5}+\left(-1/2\sqrt{-3}+1/2\right)g_{-6}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}288\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{21}_1$
h-characteristic: (0, 2, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 42
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $2B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_3+A^{1}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_3+B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{6\omega_{1}}+3V_{4\omega_{1}}+12V_{3\omega_{1}}+6V_{2\omega_{1}}+6V_{\omega_{1}}+22V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&9h_{8}+18h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{58}+4g_{52}+g_{44}+3g_{10}+3g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-44}+g_{-52}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&9h_{8}+18h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{58}+8g_{52}+2g_{44}+6g_{10}+6g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-44}-2g_{-52}-2g_{-58}\end{array}$
Centralizer type: $C_3$
Unfold the hidden panel for more information.

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$A^{20}_1$
h-characteristic: (1, 0, 1, 0, 0, 1, 0, 0)
Length of the weight dual to h: 40
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $A^{2}_2+B^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_2+B^{1}_2+A^{2}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_3+A^{2}_2+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+2V_{5\omega_{1}}+5V_{4\omega_{1}}+8V_{3\omega_{1}}+11V_{2\omega_{1}}+10V_{\omega_{1}}+7V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&10h_{8}+20h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&4g_{52}+2g_{46}+g_{44}+g_{34}+2g_{25}+3g_{16}\\ f&=&g_{-16}+g_{-25}+g_{-34}+g_{-44}+g_{-46}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&10h_{8}+20h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&8g_{52}+4g_{46}+2g_{44}+2g_{34}+4g_{25}+6g_{16}\\ [h, f]&=&-2g_{-16}-2g_{-25}-2g_{-34}-2g_{-44}-2g_{-46}-2g_{-52}\end{array}$
Centralizer type: $A^{3}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 7): $g_{-8}$, $h_{7}-h_{2}$, $h_{8}$, $g_{2}+g_{-22}$, $g_{5}-g_{-5}$, $g_{8}$, $g_{22}+g_{-2}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{7}-h_{2}$, $-h_{8}$
Cartan of centralizer (dimension: 3): $g_{5}-g_{-5}$, $-h_{8}$, $h_{7}-h_{2}$
Cartan-generating semisimple element: $-h_{8}+h_{7}-h_{2}$
adjoint action: $\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^7-10x^5+9x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -3)(x -1)(x +1)(x +3)
Eigenvalues of ad H: $0$, $3$, $1$, $-1$, $-3$
7 eigenvectors of ad H: $(0,1,0,0,0,0,0)$, $(0,0,1,0,0,0,0)$, $(0,0,0,0,1,0,0)$, $(1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1)$, $(0,0,0,1,0,0,0)$, $(0,0,0,0,0,1,0)$
Centralizer type: A^{3}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}-2h_{2}$
matching e: $g_{22}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}-2h_{2}$
matching e: $g_{22}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{20}_1$
h-characteristic: (0, 2, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 40
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $2B^{1}_2$ Containing regular semisimple subalgebra number 2: $A^{2}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{6\omega_{1}}+V_{4\omega_{1}}+16V_{3\omega_{1}}+3V_{2\omega_{1}}+37V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+16h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{58}+4g_{52}+3g_{10}+3g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-52}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+16h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{58}+8g_{52}+6g_{10}+6g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-52}-2g_{-58}\end{array}$
Centralizer type: $C_4$
Unfold the hidden panel for more information.

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$A^{19}_1$
h-characteristic: (1, 0, 1, 0, 1, 0, 0, 0)
Length of the weight dual to h: 38
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{2}_2+B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_3+A^{2}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+2V_{5\omega_{1}}+4V_{4\omega_{1}}+8V_{3\omega_{1}}+12V_{2\omega_{1}}+8V_{\omega_{1}}+13V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&9h_{8}+18h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&4g_{52}+2g_{50}+g_{44}+2g_{18}+3g_{16}\\ f&=&g_{-16}+g_{-18}+g_{-44}+g_{-50}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&9h_{8}+18h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&8g_{52}+4g_{50}+2g_{44}+4g_{18}+6g_{16}\\ [h, f]&=&-2g_{-16}-2g_{-18}-2g_{-44}-2g_{-50}-2g_{-52}\end{array}$
Centralizer type: $B_2+A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 13): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $h_{6}-h_{2}$, $h_{7}$, $h_{8}$, $g_{2}+g_{-34}$, $g_{7}$, $g_{8}$, $g_{15}$, $g_{22}$, $g_{34}+g_{-2}$
Basis of centralizer intersected with cartan (dimension: 3): $h_{6}-h_{2}$, $-h_{8}$, $-h_{7}$
Cartan of centralizer (dimension: 3): $-h_{8}$, $-h_{7}$, $h_{6}-h_{2}$
Cartan-generating semisimple element: $-h_{8}-h_{7}+h_{6}-h_{2}$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{13}-17/2x^{11}+393/16x^9-239/8x^7+241/16x^5-9/4x^3$
Factorization of characteristic polynomial of ad H: 1/16(x )(x )(x )(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(2x -3)(2x -1)(2x +1)(2x +3)
Eigenvalues of ad H: $0$, $2$, $1$, $-1$, $-2$, $3/2$, $1/2$, $-1/2$, $-3/2$
13 eigenvectors of ad H: $(0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0)$
Centralizer type: B^{1}_2+A^{3}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9 & -1/18\\ -1/18 & 1/18\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 0), (-1, 0), (2, 1), (-2, -1), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36 & -36\\ -36 & 72\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}+2h_{6}-2h_{2}$
matching e: $g_{34}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}+2h_{6}-2h_{2}$
matching e: $g_{34}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{18}_1$
h-characteristic: (1, 0, 1, 1, 0, 0, 0, 0)
Length of the weight dual to h: 36
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{2}_2+B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+2V_{5\omega_{1}}+3V_{4\omega_{1}}+8V_{3\omega_{1}}+14V_{2\omega_{1}}+2V_{\omega_{1}}+24V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+16h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&2g_{54}+4g_{52}+3g_{16}+2g_{11}\\ f&=&g_{-11}+g_{-16}+g_{-52}+g_{-54}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+16h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&4g_{54}+8g_{52}+6g_{16}+4g_{11}\\ [h, f]&=&-2g_{-11}-2g_{-16}-2g_{-52}-2g_{-54}\end{array}$
Centralizer type: $C_3+A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 24): $g_{-34}$, $g_{-28}$, $g_{-22}$, $g_{-21}$, $g_{-15}$, $g_{-14}$, $g_{-8}$, $g_{-7}$, $g_{-6}$, $h_{5}-h_{2}$, $h_{6}$, $h_{7}$, $h_{8}$, $g_{2}+g_{-44}$, $g_{6}$, $g_{7}$, $g_{8}$, $g_{14}$, $g_{15}$, $g_{21}$, $g_{22}$, $g_{28}$, $g_{34}$, $g_{44}+g_{-2}$
Basis of centralizer intersected with cartan (dimension: 4): $h_{5}-h_{2}$, $-h_{8}$, $-h_{7}$, $-h_{6}$
Cartan of centralizer (dimension: 4): $-h_{8}$, $-h_{7}$, $-h_{6}$, $h_{5}-h_{2}$
Cartan-generating semisimple element: $-9h_{8}+7h_{7}+5h_{6}+h_{5}-h_{2}$
adjoint action: $\begin{pmatrix}-4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 21/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 23/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -29/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 25 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -27/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 27/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -25 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 29/2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -23/2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -21/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{24}-1291x^{22}+4674543/8x^{20}-1997462351/16x^{18}+3441686272361/256x^{16}-11049787068411/16x^{14}+1786351718615063/128x^{12}-7476599047022629/64x^{10}+101274541816996881/256x^8-31564288362182949/64x^6+804526423430625/4x^4$
Factorization of characteristic polynomial of ad H: 1/256(x )(x )(x )(x )(x -25)(x -4)(x -3)(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(x +3)(x +4)(x +25)(2x -29)(2x -27)(2x -23)(2x -21)(2x +21)(2x +23)(2x +27)(2x +29)
Eigenvalues of ad H: $0$, $25$, $4$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-4$, $-25$, $29/2$, $27/2$, $23/2$, $21/2$, $-21/2$, $-23/2$, $-27/2$, $-29/2$
24 eigenvectors of ad H: $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: C^{1}_3+A^{3}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/18 & -1/36 & -1/18\\ -1/36 & 1/18 & 0\\ -1/18 & 0 & 1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (3 total):
$2h_{6}$
matching e: $g_{6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{6}$
matching e: $g_{6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (1, 0, 1), (-1, 0, -1), (1, 0, 2), (-1, 0, -2), (0, 0, 1), (0, 0, -1), (1, 0, 0), (-1, 0, 0), (2, 1, 2), (-2, -1, -2), (1, 1, 2), (-1, -1, -2), (1, 1, 0), (-1, -1, 0), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}72 & -36 & -36\\ -36 & 72 & 0\\ -36 & 0 & 36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}+2h_{6}+2h_{5}-2h_{2}$
matching e: $g_{44}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}+2h_{6}+2h_{5}-2h_{2}$
matching e: $g_{44}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{18}_1$
h-characteristic: (0, 0, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 36
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $2A^{2}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $2A^{2}_2+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $10V_{4\omega_{1}}+8V_{3\omega_{1}}+9V_{2\omega_{1}}+8V_{\omega_{1}}+11V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&10h_{8}+20h_{7}+20h_{6}+18h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ e&=&2g_{49}+g_{44}+2g_{37}+g_{34}+2g_{30}+2g_{26}\\ f&=&g_{-26}+g_{-30}+g_{-34}+g_{-37}+g_{-44}+g_{-49}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&10h_{8}+20h_{7}+20h_{6}+18h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&4g_{49}+2g_{44}+4g_{37}+2g_{34}+4g_{30}+4g_{26}\\ [h, f]&=&-2g_{-26}-2g_{-30}-2g_{-34}-2g_{-37}-2g_{-44}-2g_{-49}\end{array}$
Centralizer type: $B^{3}_2$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 11): $h_{7}+h_{3}-h_{1}$, $h_{8}-2h_{3}$, $g_{1}+g_{-22}$, $g_{3}+g_{-8}$, $g_{5}-g_{-5}$, $g_{8}+g_{-3}$, $g_{9}-g_{-7}-g_{-10}$, $g_{10}+g_{7}-g_{-9}$, $g_{15}+g_{2}+g_{-16}$, $g_{16}+g_{-2}+g_{-15}$, $g_{22}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 2): $1/2h_{8}-h_{3}$, $1/2h_{8}+h_{7}-h_{1}$
Cartan of centralizer (dimension: 3): $1/2h_{8}-h_{3}$, $1/2h_{8}+h_{7}-h_{1}$, $g_{5}-g_{-5}$
Cartan-generating semisimple element: $5h_{8}+9h_{7}-h_{3}-9h_{1}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{11}-123x^9+3843x^7-36121x^5+32400x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -9)(x -5)(x -4)(x -1)(x +1)(x +4)(x +5)(x +9)
Eigenvalues of ad H: $0$, $9$, $5$, $4$, $1$, $-1$, $-4$, $-5$, $-9$
11 eigenvectors of ad H: $(1,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,1,0,0,0,0,0,0,0,0)$
Centralizer type: B^{3}_2
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/54 & -1/54\\ -1/54 & 1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$2h_{7}+2h_{3}-2h_{1}$
matching e: $g_{10}+g_{7}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$h_{8}-2h_{3}$
matching e: $g_{8}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{7}+2h_{3}-2h_{1}$
matching e: $g_{10}+g_{7}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$h_{8}-2h_{3}$
matching e: $g_{8}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}216 & -108\\ -108 & 108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{17}_1$
h-characteristic: (0, 0, 0, 1, 1, 0, 0, 0)
Length of the weight dual to h: 34
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $2A^{2}_2+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $10V_{4\omega_{1}}+4V_{3\omega_{1}}+15V_{2\omega_{1}}+6V_{\omega_{1}}+13V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&9h_{8}+18h_{7}+18h_{6}+18h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ e&=&2g_{53}+g_{44}+2g_{42}+2g_{24}+2g_{19}\\ f&=&g_{-19}+g_{-24}+g_{-42}+g_{-44}+g_{-53}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&9h_{8}+18h_{7}+18h_{6}+18h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&4g_{53}+2g_{44}+4g_{42}+4g_{24}+4g_{19}\\ [h, f]&=&-2g_{-19}-2g_{-24}-2g_{-42}-2g_{-44}-2g_{-53}\end{array}$
Centralizer type: $B^{3}_2+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 13): $g_{-8}$, $h_{6}+h_{3}-h_{1}$, $h_{7}-h_{3}$, $h_{8}$, $g_{1}+g_{-34}$, $g_{3}+g_{-22}$, $g_{8}$, $g_{9}-g_{-6}-g_{-10}$, $g_{10}+g_{6}-g_{-9}$, $g_{16}+g_{-2}+g_{-28}$, $g_{22}+g_{-3}$, $g_{28}+g_{2}+g_{-16}$, $g_{34}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 3): $h_{7}+h_{6}-h_{1}$, $-h_{8}$, $h_{7}-h_{3}$
Cartan of centralizer (dimension: 3): $g_{34}+g_{28}+g_{22}+g_{16}+g_{10}+g_{9}-g_{8}+g_{6}+g_{3}+g_{2}+g_{1}+h_{7}+h_{6}-h_{1}+g_{-1}+g_{-2}+g_{-3}-g_{-6}-g_{-8}-g_{-9}-g_{-10}+g_{-16}+g_{-22}+g_{-28}+g_{-34}$, $-2g_{34}-2g_{28}-2g_{22}-2g_{16}-2g_{10}-2g_{9}-2g_{6}-2g_{3}-2g_{2}-2g_{1}-h_{8}-2h_{7}-2h_{6}+2h_{1}-2g_{-1}-2g_{-2}-2g_{-3}+2g_{-6}+2g_{-9}+2g_{-10}-2g_{-16}-2g_{-22}-2g_{-28}-2g_{-34}$, $-g_{34}-9/4g_{28}-3/8g_{22}-7/4g_{16}-9/4g_{10}-7/4g_{9}-9/4g_{6}-11/8g_{3}-9/4g_{2}-g_{1}-h_{6}-h_{3}+h_{1}-g_{-1}-7/4g_{-2}-3/8g_{-3}+7/4g_{-6}+9/4g_{-9}+7/4g_{-10}-9/4g_{-16}-11/8g_{-22}-7/4g_{-28}-g_{-34}$
Cartan-generating semisimple element: $g_{34}+g_{28}+g_{22}+g_{16}+g_{10}+g_{9}+g_{8}+g_{6}+g_{3}+g_{2}+g_{1}+h_{8}+h_{7}+h_{6}-h_{1}+g_{-1}+g_{-2}+g_{-3}-g_{-6}+g_{-8}-g_{-9}-g_{-10}+g_{-16}+g_{-22}+g_{-28}+g_{-34}$
adjoint action: $\begin{pmatrix}-1 & 0 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & -2 & 2 & 2 & 0 & -2 & -2\\ 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 4 & -2 & -4 & -2\\ 1 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 2 & -1 & -2 & -1\\ 0 & 1 & 0 & 0 & -1 & 0 & 0 & -2 & 0 & 2 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & -2 & 0 & 0 & 0\\ 0 & 0 & 1 & -2 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & -1/2 & 0 & -1 & 0 & 0 & -1/2 & 0 & -1 & 1 & 1 & 0\\ 0 & -1 & 1/2 & 0 & 0 & 1 & 0 & 0 & 1/2 & 1 & 0 & -1 & -1\\ 0 & 0 & 1/2 & 0 & -1 & 1 & 0 & -1 & 1 & -1/2 & 0 & 0 & 0\\ 0 & 1 & -1 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & -1/2 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 1 & 1/2 & -1\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 2 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{13}-37/2x^{11}+2049/16x^9-3147/8x^7+7265/16x^5-25/4x^3$
Factorization of characteristic polynomial of ad H: 1/16(x )(x )(x )(x -2)(x +2)(x^2-5)(x^2-5)(4x^2-8x -1)(4x^2+8x -1)
Eigenvalues of ad H: $0$, $2$, $-2$, $\sqrt{5}$, $-\sqrt{5}$, $1/2\sqrt{5}+1$, $-1/2\sqrt{5}+1$, $1/2\sqrt{5}-1$, $-1/2\sqrt{5}-1$
13 eigenvectors of ad H: $(1,0,0,12,0,0,1,0,0,0,0,0,0)$, $(0,0,−45,−25,0,310,0,35,1,35,−12,1,0)$, $(0,1,95,910,1,710,0,25,0,25,32,0,1)$, $(0,−65,−98,−916,−925,−9400,0,−320,−320,−9100,116,−14,1)$, $(0,0,2,1,0,−1,0,0,0,0,1,0,0)$, $(125−32,0,0,−125+12,0,0,1,0,0,0,0,0,0)$, $(0,−5+1,−2215+493,−22125+4932,125−32,1105−246,0,85−18,−55+11,85−18,−1105+246,−55+11,1)$, $(−125−32,0,0,125+12,0,0,1,0,0,0,0,0,0)$, $(0,5+1,2215+493,22125+4932,−125−32,−1105−246,0,−85−18,55+11,−85−18,1105+246,55+11,1)$, $(0,−125−110,325−92,345−94,−3105+310,−345+3320,0,−25165+26980,−525+21740,−123805+26780,545−114,−525+438,1)$, $(0,125−110,−325−92,−345−94,3105+310,345+3320,0,25165+26980,525+21740,123805+26780,−545−114,525+438,1)$, $(0,0,−205+44,−105+22,0,105−22,0,125−12,1,125−12,−105+22,1,0)$, $(0,0,205+44,105+22,0,−105−22,0,−125−12,1,−125−12,105+22,1,0)$
Centralizer type: B^{3}_2+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/27 & -1/54\\ -1/54 & 1/54\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$-10g_{28}+5g_{22}-6g_{16}-10g_{10}-6g_{9}-10g_{6}-3g_{3}-10g_{2}+4h_{8}+8h_{7}-8h_{3}-6g_{-2}+5g_{-3}+6g_{-6}+10g_{-9}+6g_{-10}-10g_{-16}-3g_{-22}-6g_{-28}$
matching e: $g_{34}-1/4g_{28}+1/16g_{22}-9/100g_{16}-3/20g_{10}-3/20g_{9}-3/20g_{6}-9/400g_{3}-1/4g_{2}-9/25g_{1}-9/16h_{8}-9/8h_{7}-6/5h_{6}-3/40h_{3}+6/5h_{1}+g_{-1}-9/100g_{-2}+1/16g_{-3}+3/20g_{-6}+3/20g_{-9}+3/20g_{-10}-1/4g_{-16}-9/400g_{-22}-9/100g_{-28}-9/25g_{-34}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 20 & -12 & -20 & 0 & 12 & 0\\ 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & -40 & 6 & 24 & 0\\ 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & -20 & 3 & 12 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 12 & 0 & -12 & 0 & 0 & 0\\ 0 & 3 & -3 & 0 & 0 & -8 & 0 & 0 & -12 & 20 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -6 & 3 & 0 & 10 & 0 & 0 & 4 & 0 & -5 & -6 & 0 & 0\\ 0 & 10 & -5 & 0 & 0 & -10 & 0 & 0 & -4 & 0 & 0 & 3 & 6\\ 0 & 0 & -3 & 0 & 10 & -6 & 0 & 3 & 0 & -4 & 0 & 0 & 0\\ 0 & 5 & -5 & 0 & 0 & 0 & 0 & -20 & 0 & 0 & 8 & 12 & 0\\ 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & -5 & 0 & -10 & 4 & 6\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 20 & 0 & 0 & -20 & 0\\ \end{pmatrix}$
$2/5\sqrt{5}g_{34}+\left(22/5\sqrt{5}+10\right)g_{28}+\left(-8/5\sqrt{5}-5\right)g_{22}+\left(14/5\sqrt{5}+6\right)g_{16}+\left(22/5\sqrt{5}+10\right)g_{10}+\left(14/5\sqrt{5}+6\right)g_{9}+\left(22/5\sqrt{5}+10\right)g_{6}+\left(8/5\sqrt{5}+3\right)g_{3}+\left(22/5\sqrt{5}+10\right)g_{2}+2/5\sqrt{5}g_{1}+\left(-7/5\sqrt{5}-4\right)h_{8}+\left(-14/5\sqrt{5}-8\right)h_{7}+2/5\sqrt{5}h_{6}+\left(16/5\sqrt{5}+8\right)h_{3}-2/5\sqrt{5}h_{1}+2/5\sqrt{5}g_{-1}+\left(14/5\sqrt{5}+6\right)g_{-2}+\left(-8/5\sqrt{5}-5\right)g_{-3}+\left(-14/5\sqrt{5}-6\right)g_{-6}+\left(-22/5\sqrt{5}-10\right)g_{-9}+\left(-14/5\sqrt{5}-6\right)g_{-10}+\left(22/5\sqrt{5}+10\right)g_{-16}+\left(8/5\sqrt{5}+3\right)g_{-22}+\left(14/5\sqrt{5}+6\right)g_{-28}+2/5\sqrt{5}g_{-34}$
matching e: $g_{28}+\left(-10\sqrt{5}+22\right)g_{22}+\left(1/2\sqrt{5}-1/2\right)g_{16}+g_{10}+\left(1/2\sqrt{5}-1/2\right)g_{9}+g_{6}+\left(10\sqrt{5}-22\right)g_{3}+g_{2}+\left(-10\sqrt{5}+22\right)h_{8}+\left(-20\sqrt{5}+44\right)h_{7}+\left(20\sqrt{5}-44\right)h_{3}+\left(1/2\sqrt{5}-1/2\right)g_{-2}+\left(-10\sqrt{5}+22\right)g_{-3}+\left(-1/2\sqrt{5}+1/2\right)g_{-6}-g_{-9}+\left(-1/2\sqrt{5}+1/2\right)g_{-10}+g_{-16}+\left(10\sqrt{5}-22\right)g_{-22}+\left(1/2\sqrt{5}-1/2\right)g_{-28}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 4/5\sqrt{5} & 0 & 0 & -44/5\sqrt{5}-20 & 28/5\sqrt{5}+12 & 44/5\sqrt{5}+20 & 0 & -28/5\sqrt{5}-12 & -4/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 4/5\sqrt{5} & -16/5\sqrt{5}-10 & 0 & 0 & 0 & 88/5\sqrt{5}+40 & -16/5\sqrt{5}-6 & -56/5\sqrt{5}-24 & -4/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 2/5\sqrt{5} & -8/5\sqrt{5}-5 & 0 & 0 & 0 & 44/5\sqrt{5}+20 & -8/5\sqrt{5}-3 & -28/5\sqrt{5}-12 & -2/5\sqrt{5}\\ 0 & 2/5\sqrt{5} & 0 & 0 & -2/5\sqrt{5} & 0 & 0 & -28/5\sqrt{5}-12 & 0 & 28/5\sqrt{5}+12 & 0 & 0 & 0\\ 0 & -8/5\sqrt{5}-3 & 8/5\sqrt{5}+3 & 0 & 0 & 16/5\sqrt{5}+8 & 0 & 0 & 28/5\sqrt{5}+12 & -44/5\sqrt{5}-20 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 14/5\sqrt{5}+6 & -7/5\sqrt{5}-3 & 0 & -22/5\sqrt{5}-10 & 0 & 0 & -9/5\sqrt{5}-4 & 0 & 8/5\sqrt{5}+5 & 14/5\sqrt{5}+6 & 2/5\sqrt{5} & 0\\ 0 & -22/5\sqrt{5}-10 & 11/5\sqrt{5}+5 & 0 & 0 & 22/5\sqrt{5}+10 & 0 & 0 & 9/5\sqrt{5}+4 & 2/5\sqrt{5} & 0 & -8/5\sqrt{5}-3 & -14/5\sqrt{5}-6\\ 0 & 0 & 7/5\sqrt{5}+3 & 0 & -22/5\sqrt{5}-10 & 14/5\sqrt{5}+6 & 0 & -8/5\sqrt{5}-3 & 2/5\sqrt{5} & 7/5\sqrt{5}+4 & 0 & 0 & 0\\ 0 & -8/5\sqrt{5}-5 & 8/5\sqrt{5}+5 & 0 & 0 & 0 & 0 & 44/5\sqrt{5}+20 & 0 & 0 & -16/5\sqrt{5}-8 & -28/5\sqrt{5}-12 & 0\\ 0 & 0 & -11/5\sqrt{5}-5 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 8/5\sqrt{5}+5 & 0 & 22/5\sqrt{5}+10 & -7/5\sqrt{5}-4 & -14/5\sqrt{5}-6\\ 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & -44/5\sqrt{5}-20 & 0 & 0 & 44/5\sqrt{5}+20 & 2/5\sqrt{5}\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-10g_{28}+5g_{22}-6g_{16}-10g_{10}-6g_{9}-10g_{6}-3g_{3}-10g_{2}+4h_{8}+8h_{7}-8h_{3}-6g_{-2}+5g_{-3}+6g_{-6}+10g_{-9}+6g_{-10}-10g_{-16}-3g_{-22}-6g_{-28}$
matching e: $g_{34}-1/4g_{28}+1/16g_{22}-9/100g_{16}-3/20g_{10}-3/20g_{9}-3/20g_{6}-9/400g_{3}-1/4g_{2}-9/25g_{1}-9/16h_{8}-9/8h_{7}-6/5h_{6}-3/40h_{3}+6/5h_{1}+g_{-1}-9/100g_{-2}+1/16g_{-3}+3/20g_{-6}+3/20g_{-9}+3/20g_{-10}-1/4g_{-16}-9/400g_{-22}-9/100g_{-28}-9/25g_{-34}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 20 & -12 & -20 & 0 & 12 & 0\\ 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & -40 & 6 & 24 & 0\\ 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & -20 & 3 & 12 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 12 & 0 & -12 & 0 & 0 & 0\\ 0 & 3 & -3 & 0 & 0 & -8 & 0 & 0 & -12 & 20 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -6 & 3 & 0 & 10 & 0 & 0 & 4 & 0 & -5 & -6 & 0 & 0\\ 0 & 10 & -5 & 0 & 0 & -10 & 0 & 0 & -4 & 0 & 0 & 3 & 6\\ 0 & 0 & -3 & 0 & 10 & -6 & 0 & 3 & 0 & -4 & 0 & 0 & 0\\ 0 & 5 & -5 & 0 & 0 & 0 & 0 & -20 & 0 & 0 & 8 & 12 & 0\\ 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & -5 & 0 & -10 & 4 & 6\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 20 & 0 & 0 & -20 & 0\\ \end{pmatrix}$
$2/5\sqrt{5}g_{34}+\left(22/5\sqrt{5}+10\right)g_{28}+\left(-8/5\sqrt{5}-5\right)g_{22}+\left(14/5\sqrt{5}+6\right)g_{16}+\left(22/5\sqrt{5}+10\right)g_{10}+\left(14/5\sqrt{5}+6\right)g_{9}+\left(22/5\sqrt{5}+10\right)g_{6}+\left(8/5\sqrt{5}+3\right)g_{3}+\left(22/5\sqrt{5}+10\right)g_{2}+2/5\sqrt{5}g_{1}+\left(-7/5\sqrt{5}-4\right)h_{8}+\left(-14/5\sqrt{5}-8\right)h_{7}+2/5\sqrt{5}h_{6}+\left(16/5\sqrt{5}+8\right)h_{3}-2/5\sqrt{5}h_{1}+2/5\sqrt{5}g_{-1}+\left(14/5\sqrt{5}+6\right)g_{-2}+\left(-8/5\sqrt{5}-5\right)g_{-3}+\left(-14/5\sqrt{5}-6\right)g_{-6}+\left(-22/5\sqrt{5}-10\right)g_{-9}+\left(-14/5\sqrt{5}-6\right)g_{-10}+\left(22/5\sqrt{5}+10\right)g_{-16}+\left(8/5\sqrt{5}+3\right)g_{-22}+\left(14/5\sqrt{5}+6\right)g_{-28}+2/5\sqrt{5}g_{-34}$
matching e: $g_{28}+\left(-10\sqrt{5}+22\right)g_{22}+\left(1/2\sqrt{5}-1/2\right)g_{16}+g_{10}+\left(1/2\sqrt{5}-1/2\right)g_{9}+g_{6}+\left(10\sqrt{5}-22\right)g_{3}+g_{2}+\left(-10\sqrt{5}+22\right)h_{8}+\left(-20\sqrt{5}+44\right)h_{7}+\left(20\sqrt{5}-44\right)h_{3}+\left(1/2\sqrt{5}-1/2\right)g_{-2}+\left(-10\sqrt{5}+22\right)g_{-3}+\left(-1/2\sqrt{5}+1/2\right)g_{-6}-g_{-9}+\left(-1/2\sqrt{5}+1/2\right)g_{-10}+g_{-16}+\left(10\sqrt{5}-22\right)g_{-22}+\left(1/2\sqrt{5}-1/2\right)g_{-28}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 4/5\sqrt{5} & 0 & 0 & -44/5\sqrt{5}-20 & 28/5\sqrt{5}+12 & 44/5\sqrt{5}+20 & 0 & -28/5\sqrt{5}-12 & -4/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 4/5\sqrt{5} & -16/5\sqrt{5}-10 & 0 & 0 & 0 & 88/5\sqrt{5}+40 & -16/5\sqrt{5}-6 & -56/5\sqrt{5}-24 & -4/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 2/5\sqrt{5} & -8/5\sqrt{5}-5 & 0 & 0 & 0 & 44/5\sqrt{5}+20 & -8/5\sqrt{5}-3 & -28/5\sqrt{5}-12 & -2/5\sqrt{5}\\ 0 & 2/5\sqrt{5} & 0 & 0 & -2/5\sqrt{5} & 0 & 0 & -28/5\sqrt{5}-12 & 0 & 28/5\sqrt{5}+12 & 0 & 0 & 0\\ 0 & -8/5\sqrt{5}-3 & 8/5\sqrt{5}+3 & 0 & 0 & 16/5\sqrt{5}+8 & 0 & 0 & 28/5\sqrt{5}+12 & -44/5\sqrt{5}-20 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 14/5\sqrt{5}+6 & -7/5\sqrt{5}-3 & 0 & -22/5\sqrt{5}-10 & 0 & 0 & -9/5\sqrt{5}-4 & 0 & 8/5\sqrt{5}+5 & 14/5\sqrt{5}+6 & 2/5\sqrt{5} & 0\\ 0 & -22/5\sqrt{5}-10 & 11/5\sqrt{5}+5 & 0 & 0 & 22/5\sqrt{5}+10 & 0 & 0 & 9/5\sqrt{5}+4 & 2/5\sqrt{5} & 0 & -8/5\sqrt{5}-3 & -14/5\sqrt{5}-6\\ 0 & 0 & 7/5\sqrt{5}+3 & 0 & -22/5\sqrt{5}-10 & 14/5\sqrt{5}+6 & 0 & -8/5\sqrt{5}-3 & 2/5\sqrt{5} & 7/5\sqrt{5}+4 & 0 & 0 & 0\\ 0 & -8/5\sqrt{5}-5 & 8/5\sqrt{5}+5 & 0 & 0 & 0 & 0 & 44/5\sqrt{5}+20 & 0 & 0 & -16/5\sqrt{5}-8 & -28/5\sqrt{5}-12 & 0\\ 0 & 0 & -11/5\sqrt{5}-5 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 8/5\sqrt{5}+5 & 0 & 22/5\sqrt{5}+10 & -7/5\sqrt{5}-4 & -14/5\sqrt{5}-6\\ 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & -44/5\sqrt{5}-20 & 0 & 0 & 44/5\sqrt{5}+20 & 2/5\sqrt{5}\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 0), (-1, 0), (1, 1), (-1, -1), (2, 1), (-2, -1), (0, -1), (0, 1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108 & -108\\ -108 & 216\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$2/5\sqrt{5}g_{8}+1/5\sqrt{5}h_{8}+2/5\sqrt{5}g_{-8}$
matching e: $g_{8}+\left(-1/2\sqrt{5}+1/2\right)h_{8}+\left(1/2\sqrt{5}-3/2\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2/5\sqrt{5} & 0 & -2/5\sqrt{5} & 4/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/5\sqrt{5} & -4/5\sqrt{5} & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2/5\sqrt{5}g_{8}+1/5\sqrt{5}h_{8}+2/5\sqrt{5}g_{-8}$
matching e: $g_{8}+\left(-1/2\sqrt{5}+1/2\right)h_{8}+\left(1/2\sqrt{5}-3/2\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2/5\sqrt{5} & 0 & -2/5\sqrt{5} & 4/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/5\sqrt{5} & -4/5\sqrt{5} & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{16}_1$
h-characteristic: (2, 0, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 32
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 7
Containing regular semisimple subalgebra number 1: $B^{1}_2+2A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_2+3A^{2}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_3+5A^{1}_1$ Containing regular semisimple subalgebra number 4: $B^{1}_2+6A^{1}_1$ Containing regular semisimple subalgebra number 5: $C^{1}_3+A^{2}_1+3A^{1}_1$ Containing regular semisimple subalgebra number 6: $B^{1}_2+A^{2}_1+4A^{1}_1$ Containing regular semisimple subalgebra number 7: $C^{1}_3+2A^{2}_1+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+6V_{4\omega_{1}}+28V_{2\omega_{1}}+15V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&10h_{8}+18h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&4g_{62}+g_{55}+g_{39}+g_{22}+g_{8}+3g_{1}\\ f&=&g_{-1}+g_{-8}+g_{-22}+g_{-39}+g_{-55}+g_{-62}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&10h_{8}+18h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&8g_{62}+2g_{55}+2g_{39}+2g_{22}+2g_{8}+6g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-8}-2g_{-22}-2g_{-39}-2g_{-55}-2g_{-62}\end{array}$
Centralizer type: $A^{4}_3$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 15): $h_{3}$, $h_{5}$, $g_{4}-g_{-18}$, $g_{6}-g_{-13}$, $g_{7}-g_{-7}$, $g_{11}-g_{-12}$, $g_{12}-g_{-11}$, $g_{13}-g_{-6}$, $g_{14}-g_{-20}$, $g_{18}-g_{-4}$, $g_{19}-g_{-25}$, $g_{20}-g_{-14}$, $g_{25}-g_{-19}$, $g_{26}-g_{-31}$, $g_{31}-g_{-26}$
Basis of centralizer intersected with cartan (dimension: 2): $-h_{5}$, $-h_{3}$
Cartan of centralizer (dimension: 3): $-h_{5}$, $-h_{3}$, $g_{7}-g_{-7}$
Cartan-generating semisimple element: $-h_{5}-9h_{3}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9/2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{15}-82x^{13}+20091/8x^{11}-548047/16x^9+46255537/256x^7-21523401/256x^5+164025/16x^3$
Factorization of characteristic polynomial of ad H: 1/256(x )(x )(x )(x -5)(x -4)(x +4)(x +5)(2x -9)(2x -9)(2x -1)(2x -1)(2x +1)(2x +1)(2x +9)(2x +9)
Eigenvalues of ad H: $0$, $5$, $4$, $-4$, $-5$, $9/2$, $1/2$, $-1/2$, $-9/2$
15 eigenvectors of ad H: $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$
Centralizer type: A^{4}_3
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/36 & 0 & -1/72\\ 0 & 1/36 & -1/72\\ -1/72 & -1/72 & 1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (3 total):
$\sqrt{-1}g_{7}-2h_{5}-\sqrt{-1}g_{-7}$
matching e: $g_{14}+\sqrt{-1}g_{6}-\sqrt{-1}g_{-13}-g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0\\ 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0\\ \end{pmatrix}$
$-\sqrt{-1}g_{7}-2h_{5}+\sqrt{-1}g_{-7}$
matching e: $g_{14}-\sqrt{-1}g_{6}+\sqrt{-1}g_{-13}-g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0\\ 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0\\ \end{pmatrix}$
$2h_{5}-2h_{3}$
matching e: $g_{12}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$\sqrt{-1}g_{7}-2h_{5}-\sqrt{-1}g_{-7}$
matching e: $g_{14}+\sqrt{-1}g_{6}-\sqrt{-1}g_{-13}-g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0\\ 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0\\ \end{pmatrix}$
$-\sqrt{-1}g_{7}-2h_{5}+\sqrt{-1}g_{-7}$
matching e: $g_{14}-\sqrt{-1}g_{6}+\sqrt{-1}g_{-13}-g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0\\ 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0\\ \end{pmatrix}$
$2h_{5}-2h_{3}$
matching e: $g_{12}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (0, 0, 1), (0, 0, -1), (1, 0, 1), (-1, 0, -1), (0, 1, 1), (0, -1, -1), (1, 0, 0), (-1, 0, 0), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144 & 0 & -72\\ 0 & 144 & -72\\ -72 & -72 & 144\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{16}_1$
h-characteristic: (0, 0, 0, 2, 0, 0, 0, 0)
Length of the weight dual to h: 32
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $2A^{2}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $10V_{4\omega_{1}}+22V_{2\omega_{1}}+20V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+16h_{7}+16h_{6}+16h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ e&=&2g_{56}+2g_{47}+2g_{17}+2g_{12}\\ f&=&g_{-12}+g_{-17}+g_{-47}+g_{-56}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+16h_{7}+16h_{6}+16h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&4g_{56}+4g_{47}+4g_{17}+4g_{12}\\ [h, f]&=&-2g_{-12}-2g_{-17}-2g_{-47}-2g_{-56}\end{array}$
Centralizer type: $B^{3}_2+B_2$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 20): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $h_{5}+h_{3}-h_{1}$, $h_{6}-h_{3}$, $h_{7}$, $h_{8}$, $g_{1}+g_{-44}$, $g_{3}+g_{-34}$, $g_{7}$, $g_{8}$, $g_{9}-g_{-5}-g_{-10}$, $g_{10}+g_{5}-g_{-9}$, $g_{15}$, $g_{16}+g_{-2}+g_{-39}$, $g_{22}$, $g_{34}+g_{-3}$, $g_{39}+g_{2}+g_{-16}$, $g_{44}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 4): $-h_{8}$, $-h_{7}$, $h_{6}-h_{3}$, $h_{6}+h_{5}-h_{1}$
Cartan of centralizer (dimension: 4): $h_{6}+h_{5}-h_{1}$, $h_{6}-h_{3}$, $-h_{7}$, $-h_{8}$
Cartan-generating semisimple element: $-h_{8}-9h_{7}-12h_{6}-5h_{5}+7h_{3}+5h_{1}$
adjoint action: $\begin{pmatrix}-3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{20}-198x^{18}+15879x^{16}-661244x^{14}+15216399x^{12}-189748758x^{10}+1166457721x^8-2936089800x^6+1944810000x^4$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -7)(x -7)(x -6)(x -5)(x -5)(x -3)(x -2)(x -1)(x +1)(x +2)(x +3)(x +5)(x +5)(x +6)(x +7)(x +7)
Eigenvalues of ad H: $0$, $7$, $6$, $5$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-5$, $-6$, $-7$
20 eigenvectors of ad H: $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$
Centralizer type: B^{3}_2+B^{1}_2
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/27 & -1/54\\ -1/54 & 1/54\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$-h_{8}-2h_{7}-2h_{6}-2h_{5}+2h_{1}$
matching e: $g_{1}+g_{-44}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
$2h_{5}+2h_{3}-2h_{1}$
matching e: $g_{10}+g_{5}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}-2h_{7}-2h_{6}-2h_{5}+2h_{1}$
matching e: $g_{1}+g_{-44}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
$2h_{5}+2h_{3}-2h_{1}$
matching e: $g_{10}+g_{5}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (2, 1), (-2, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108 & -108\\ -108 & 216\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/9 & -1/18\\ -1/18 & 1/18\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (2, 1), (-2, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36 & -36\\ -36 & 72\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{15}_1$
h-characteristic: (2, 0, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 30
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 6
Containing regular semisimple subalgebra number 1: $B^{1}_2+2A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_3+4A^{1}_1$ Containing regular semisimple subalgebra number 3: $B^{1}_2+5A^{1}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_3+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 5: $B^{1}_2+A^{2}_1+3A^{1}_1$ Containing regular semisimple subalgebra number 6: $C^{1}_3+2A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+5V_{4\omega_{1}}+2V_{3\omega_{1}}+21V_{2\omega_{1}}+10V_{\omega_{1}}+13V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&9h_{8}+18h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&4g_{62}+g_{55}+g_{39}+g_{22}+3g_{1}\\ f&=&g_{-1}+g_{-22}+g_{-39}+g_{-55}+g_{-62}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&9h_{8}+18h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&8g_{62}+2g_{55}+2g_{39}+2g_{22}+6g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-22}-2g_{-39}-2g_{-55}-2g_{-62}\end{array}$
Centralizer type: $B^{4}_2+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 13): $g_{-8}$, $h_{3}$, $h_{5}$, $h_{8}$, $g_{4}-g_{-18}$, $g_{6}-g_{-13}$, $g_{8}$, $g_{11}-g_{-12}$, $g_{12}-g_{-11}$, $g_{13}-g_{-6}$, $g_{18}-g_{-4}$, $g_{19}-g_{-25}$, $g_{25}-g_{-19}$
Basis of centralizer intersected with cartan (dimension: 3): $-h_{3}$, $-h_{8}$, $-h_{5}$
Cartan of centralizer (dimension: 3): $-h_{3}$, $-h_{8}$, $-h_{5}$
Cartan-generating semisimple element: $-9h_{8}+7h_{5}-h_{3}$
adjoint action: $\begin{pmatrix}18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{13}-723/2x^{11}+201753/16x^9-2412397/16x^7+2433789/4x^5-142884x^3$
Factorization of characteristic polynomial of ad H: 1/16(x )(x )(x )(x -18)(x -4)(x -3)(x +3)(x +4)(x +18)(2x -7)(2x -1)(2x +1)(2x +7)
Eigenvalues of ad H: $0$, $18$, $4$, $3$, $-3$, $-4$, $-18$, $7/2$, $1/2$, $-1/2$, $-7/2$
13 eigenvectors of ad H: $(0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0)$
Centralizer type: B^{4}_2+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36 & -1/72\\ -1/72 & 1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$2h_{5}+2h_{3}$
matching e: $g_{18}-g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
$-4h_{3}$
matching e: $g_{19}-g_{-25}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{5}+2h_{3}$
matching e: $g_{18}-g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
$-4h_{3}$
matching e: $g_{19}-g_{-25}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 0), (-1, 0), (2, 1), (-2, -1), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144 & -144\\ -144 & 288\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{14}_1$
h-characteristic: (2, 0, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 28
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $B^{1}_2+4A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_2+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $B^{1}_2+2A^{2}_1$ Containing regular semisimple subalgebra number 4: $C^{1}_3+3A^{1}_1$ Containing regular semisimple subalgebra number 5: $C^{1}_3+A^{2}_1+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+4V_{4\omega_{1}}+4V_{3\omega_{1}}+15V_{2\omega_{1}}+16V_{\omega_{1}}+16V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+16h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&4g_{62}+g_{58}+g_{52}+g_{44}+g_{34}+3g_{1}\\ f&=&g_{-1}+g_{-34}+g_{-44}+g_{-52}+g_{-58}+g_{-62}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+16h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&8g_{62}+2g_{58}+2g_{52}+2g_{44}+2g_{34}+6g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-34}-2g_{-44}-2g_{-52}-2g_{-58}-2g_{-62}\end{array}$
Centralizer type: $B_2+2A^{4}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 16): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $h_{7}$, $h_{8}$, $g_{3}-g_{-3}$, $g_{4}-g_{-4}$, $g_{5}-g_{-5}$, $g_{7}$, $g_{8}$, $g_{11}-g_{-11}$, $g_{12}-g_{-12}$, $g_{15}$, $g_{18}-g_{-18}$, $g_{22}$
Basis of centralizer intersected with cartan (dimension: 2): $-h_{8}$, $-h_{7}$
Cartan of centralizer (dimension: 4): $-g_{22}-g_{15}-g_{8}-3h_{8}-3h_{7}-g_{-8}-g_{-15}-g_{-22}$, $g_{18}+g_{5}+g_{4}+g_{3}-g_{-3}-g_{-4}-g_{-5}-g_{-18}$, $g_{12}+g_{11}-g_{-11}-g_{-12}$, $-g_{7}+2h_{8}+2h_{7}-g_{-7}$
Cartan-generating semisimple element: $g_{22}+g_{18}+g_{15}+g_{12}+g_{11}+g_{8}+g_{7}+g_{5}+g_{4}+g_{3}+h_{8}+h_{7}-g_{-3}-g_{-4}-g_{-5}+g_{-7}+g_{-8}-g_{-11}-g_{-12}+g_{-15}-g_{-18}+g_{-22}$
adjoint action: $\begin{pmatrix}-1 & -2 & 0 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & -1 & -1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & -1 & 0 & -1 & 2 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0\\ 2 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & -2 & 0 & -2\\ 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -2 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 1 & -1 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 & -1 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & -1 & 0 & -1 & 0\\ 0 & 1 & 1 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1\\ 0 & 0 & 0 & -2 & 1 & -2 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 & 0 & -1 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 0 & -1 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 2 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{16}-27x^{14}-37x^{12}+2279x^{10}-1464x^8-29552x^6+28800x^4$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -5)(x -3)(x -2)(x -1)(x +1)(x +2)(x +3)(x +5)(x^2+4)(x^2+8)
Eigenvalues of ad H: $0$, $5$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-5$, $2\sqrt{-1}$, $-2\sqrt{-1}$, $2\sqrt{-2}$, $-2\sqrt{-2}$
16 eigenvectors of ad H: $(0,0,0,1,−2,−2,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0)$, $(0,0,0,0,0,0,1,1,1,0,0,0,0,0,1,0)$, $(1,1,1,0,3,3,0,0,0,0,1,0,0,1,0,1)$, $(−14,−14,−14,−12,−1,−1,0,0,0,−12,1,0,0,1,0,1)$, $(−1,0,1,−1,−2,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,14,−12,0,0,0,0,−14,−1,0,0,0,0,1)$, $(1,−1,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,−1,0,1)$, $(4,0,−4,−1,−2,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,−1,2,0,0,0,0,1,−1,0,0,0,0,1)$, $(−4,−4,−4,2,4,4,0,0,0,2,1,0,0,1,0,1)$, $(0,0,0,0,0,0,−−1,−1,−1,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,−1,−1,−−1,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,−1,1,−1,0,0,−−2,−2,0,1,0)$, $(0,0,0,0,0,0,−1,1,−1,0,0,−2,−−2,0,1,0)$
Centralizer type: B^{1}_2+2A^{4}_1
Reductive components (3 total):
Scalar product computed: $\begin{pmatrix}1/18 & -1/18\\ -1/18 & 1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$2/5g_{22}+2/5g_{15}+2/5g_{8}-1/5g_{7}+8/5h_{8}+8/5h_{7}-1/5g_{-7}+2/5g_{-8}+2/5g_{-15}+2/5g_{-22}$
matching e: $g_{22}-g_{8}-1/4g_{7}-1/2h_{7}+1/4g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-8/5 & 2/5 & 0 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/5 & -8/5 & 1/5 & 2/5 & 0 & 2/5 & 0 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & -8/5 & 0 & -2/5 & 4/5 & 0 & 0 & 0 & 4/5 & 0 & 0 & 0 & 0 & 0 & 0\\ 2/5 & 2/5 & 0 & 0 & -1/5 & 1/5 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & -2/5 & 0 & 0\\ 4/5 & 4/5 & 0 & -2/5 & 0 & 0 & 0 & 0 & 0 & 2/5 & 0 & 0 & 0 & -4/5 & 0 & -4/5\\ 2/5 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & -4/5 & 0 & -2/5\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & 2/5 & 0 & 1/5 & -1/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & -2/5\\ 0 & 0 & 0 & -4/5 & 2/5 & -4/5 & 0 & 0 & 0 & 0 & 8/5 & 0 & 0 & -2/5 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0 & 0 & 0 & -2/5 & -1/5 & 0 & 0 & 8/5 & 0 & -1/5\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 0 & -4/5 & 0 & 0 & 0 & -2/5 & 0 & 8/5\\ \end{pmatrix}$
$1/2g_{7}-h_{8}-h_{7}+1/2g_{-7}$
matching e: $g_{-8}-g_{-15}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & -1 & 0 & 1/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2/5g_{22}+2/5g_{15}+2/5g_{8}-1/5g_{7}+8/5h_{8}+8/5h_{7}-1/5g_{-7}+2/5g_{-8}+2/5g_{-15}+2/5g_{-22}$
matching e: $g_{22}-g_{8}-1/4g_{7}-1/2h_{7}+1/4g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-8/5 & 2/5 & 0 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/5 & -8/5 & 1/5 & 2/5 & 0 & 2/5 & 0 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & -8/5 & 0 & -2/5 & 4/5 & 0 & 0 & 0 & 4/5 & 0 & 0 & 0 & 0 & 0 & 0\\ 2/5 & 2/5 & 0 & 0 & -1/5 & 1/5 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & -2/5 & 0 & 0\\ 4/5 & 4/5 & 0 & -2/5 & 0 & 0 & 0 & 0 & 0 & 2/5 & 0 & 0 & 0 & -4/5 & 0 & -4/5\\ 2/5 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & -4/5 & 0 & -2/5\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & 2/5 & 0 & 1/5 & -1/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & -2/5\\ 0 & 0 & 0 & -4/5 & 2/5 & -4/5 & 0 & 0 & 0 & 0 & 8/5 & 0 & 0 & -2/5 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0 & 0 & 0 & -2/5 & -1/5 & 0 & 0 & 8/5 & 0 & -1/5\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 0 & -4/5 & 0 & 0 & 0 & -2/5 & 0 & 8/5\\ \end{pmatrix}$
$1/2g_{7}-h_{8}-h_{7}+1/2g_{-7}$
matching e: $g_{-8}-g_{-15}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & -1 & 0 & 1/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}72 & -36\\ -36 & 36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-\sqrt{-1}g_{12}-\sqrt{-1}g_{11}+\sqrt{-1}g_{-11}+\sqrt{-1}g_{-12}$
matching e: $g_{18}+\sqrt{-1}g_{5}-g_{4}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}+g_{-4}-\sqrt{-1}g_{-5}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-\sqrt{-1}g_{12}-\sqrt{-1}g_{11}+\sqrt{-1}g_{-11}+\sqrt{-1}g_{-12}$
matching e: $g_{18}+\sqrt{-1}g_{5}-g_{4}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}+g_{-4}-\sqrt{-1}g_{-5}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-1/2\sqrt{-2}g_{18}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{4}-1/2\sqrt{-2}g_{3}+1/2\sqrt{-2}g_{-3}+1/2\sqrt{-2}g_{-4}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-18}$
matching e: $g_{18}+\sqrt{-2}g_{12}-\sqrt{-2}g_{11}-g_{5}+g_{4}-g_{3}+g_{-3}-g_{-4}+g_{-5}+\sqrt{-2}g_{-11}-\sqrt{-2}g_{-12}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-1/2\sqrt{-2}g_{18}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{4}-1/2\sqrt{-2}g_{3}+1/2\sqrt{-2}g_{-3}+1/2\sqrt{-2}g_{-4}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-18}$
matching e: $g_{18}+\sqrt{-2}g_{12}-\sqrt{-2}g_{11}-g_{5}+g_{4}-g_{3}+g_{-3}-g_{-4}+g_{-5}+\sqrt{-2}g_{-11}-\sqrt{-2}g_{-12}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{13}_1$
h-characteristic: (2, 0, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 26
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $C^{1}_3+A^{2}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_3+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $B^{1}_2+3A^{1}_1$ Containing regular semisimple subalgebra number 4: $B^{1}_2+A^{2}_1+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+3V_{4\omega_{1}}+6V_{3\omega_{1}}+10V_{2\omega_{1}}+18V_{\omega_{1}}+24V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&7h_{8}+14h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&g_{60}+3/2g_{52}+5/2g_{48}+3/2g_{44}+3/2g_{23}+3/2g_{16}\\ f&=&g_{-16}+g_{-23}+g_{-48}+g_{-60}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&7h_{8}+14h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&2g_{60}+3g_{52}+5g_{48}+3g_{44}+3g_{23}+3g_{16}\\ [h, f]&=&-2g_{-16}-2g_{-23}-2g_{-48}-2g_{-60}\end{array}$
Centralizer type: $C_3+A^{8}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 24): $g_{-34}$, $g_{-28}$, $g_{-22}$, $g_{-21}$, $g_{-15}$, $g_{-14}$, $g_{-8}$, $g_{-7}$, $g_{-6}$, $h_{2}$, $h_{6}$, $h_{7}$, $h_{8}$, $g_{6}$, $g_{7}$, $g_{8}$, $g_{11}-g_{3}-g_{-10}+g_{-17}$, $g_{14}$, $g_{15}$, $g_{17}-g_{10}-g_{-3}+g_{-11}$, $g_{21}$, $g_{22}$, $g_{28}$, $g_{34}$
Basis of centralizer intersected with cartan (dimension: 4): $-h_{2}$, $-h_{8}$, $-h_{7}$, $-h_{6}$
Cartan of centralizer (dimension: 4): $-h_{6}$, $-h_{2}$, $-h_{8}$, $-h_{7}$
Cartan-generating semisimple element: $-9h_{8}+7h_{7}+5h_{6}-h_{2}$
adjoint action: $\begin{pmatrix}-5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 23/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -15 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 25 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -27/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 27/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -25 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 15 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -23/2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{24}-5233/4x^{22}+4835067/8x^{20}-4244004697/32x^{18}+3812314242269/256x^{16}-851952874070973/1024x^{14}+21094036897924399/1024x^{12}-218457258081597775/1024x^{10}+861824712384500625/1024x^8-144844948291921875/128x^6+14947354775390625/64x^4$
Factorization of characteristic polynomial of ad H: 1/1024(x )(x )(x )(x )(x -25)(x -15)(x -10)(x -5)(x -2)(x +2)(x +5)(x +10)(x +15)(x +25)(2x -27)(2x -23)(2x -7)(2x -3)(2x -1)(2x +1)(2x +3)(2x +7)(2x +23)(2x +27)
Eigenvalues of ad H: $0$, $25$, $15$, $10$, $5$, $2$, $-2$, $-5$, $-10$, $-15$, $-25$, $27/2$, $23/2$, $7/2$, $3/2$, $1/2$, $-1/2$, $-3/2$, $-7/2$, $-23/2$, $-27/2$
24 eigenvectors of ad H: $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: C^{1}_3+A^{8}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9 & -1/18 & 0\\ -1/18 & 1/18 & -1/36\\ 0 & -1/36 & 1/18\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (3 total):
$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$2h_{6}$
matching e: $g_{6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$2h_{6}$
matching e: $g_{6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (2, 2, 1), (-2, -2, -1), (0, 0, 1), (0, 0, -1), (1, 1, 0), (-1, -1, 0), (1, 0, 0), (-1, 0, 0), (2, 1, 1), (-2, -1, -1), (0, 1, 1), (0, -1, -1), (2, 1, 0), (-2, -1, 0), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36 & -36 & 0\\ -36 & 72 & -36\\ 0 & -36 & 72\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-4h_{2}$
matching e: $g_{11}-g_{3}-g_{-10}+g_{-17}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-4h_{2}$
matching e: $g_{11}-g_{3}-g_{-10}+g_{-17}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}288\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{13}_1$
h-characteristic: (0, 1, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 26
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $A^{2}_2+2A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_2+5A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{2}_2+A^{2}_1+3A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{4\omega_{1}}+10V_{3\omega_{1}}+16V_{2\omega_{1}}+10V_{\omega_{1}}+13V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&9h_{8}+18h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&g_{55}+2g_{45}+g_{39}+2g_{36}+g_{22}\\ f&=&g_{-22}+g_{-36}+g_{-39}+g_{-45}+g_{-55}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&9h_{8}+18h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&2g_{55}+4g_{45}+2g_{39}+4g_{36}+2g_{22}\\ [h, f]&=&-2g_{-22}-2g_{-36}-2g_{-39}-2g_{-45}-2g_{-55}\end{array}$
Centralizer type: $B^{4}_2+A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 13): $h_{3}$, $h_{5}$, $h_{8}-2h_{1}$, $g_{1}+g_{-8}$, $g_{4}-g_{-18}$, $g_{6}-g_{-13}$, $g_{8}+g_{-1}$, $g_{11}-g_{-12}$, $g_{12}-g_{-11}$, $g_{13}-g_{-6}$, $g_{18}-g_{-4}$, $g_{19}-g_{-25}$, $g_{25}-g_{-19}$
Basis of centralizer intersected with cartan (dimension: 3): $-h_{5}$, $-h_{3}$, $1/2h_{8}-h_{1}$
Cartan of centralizer (dimension: 3): $-h_{5}$, $-h_{3}$, $1/2h_{8}-h_{1}$
Cartan-generating semisimple element: $-7/2h_{8}-h_{5}-9h_{3}+7h_{1}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9/2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{13}-221/2x^{11}+68145/16x^9-555521/8x^7+6623929/16x^5-99225x^3$
Factorization of characteristic polynomial of ad H: 1/16(x )(x )(x )(x -7)(x -5)(x -4)(x +4)(x +5)(x +7)(2x -9)(2x -1)(2x +1)(2x +9)
Eigenvalues of ad H: $0$, $7$, $5$, $4$, $-4$, $-5$, $-7$, $9/2$, $1/2$, $-1/2$, $-9/2$
13 eigenvectors of ad H: $(1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1)$
Centralizer type: B^{4}_2+A^{3}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{8}+2h_{1}$
matching e: $g_{1}+g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}+2h_{1}$
matching e: $g_{1}+g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36 & -1/72\\ -1/72 & 1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$2h_{5}-2h_{3}$
matching e: $g_{12}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
$-4h_{5}$
matching e: $g_{6}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{5}-2h_{3}$
matching e: $g_{12}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
$-4h_{5}$
matching e: $g_{6}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 0), (-1, 0), (2, 1), (-2, -1), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144 & -144\\ -144 & 288\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{12}_1$
h-characteristic: (2, 0, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 24
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $B^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_2+A^{2}_1$ Containing regular semisimple subalgebra number 3: $C^{1}_3+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+2V_{4\omega_{1}}+8V_{3\omega_{1}}+6V_{2\omega_{1}}+16V_{\omega_{1}}+37V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&6h_{8}+12h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&4g_{62}+g_{58}+g_{52}+3g_{1}\\ f&=&g_{-1}+g_{-52}+g_{-58}+g_{-62}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&6h_{8}+12h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&8g_{62}+2g_{58}+2g_{52}+6g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-52}-2g_{-58}-2g_{-62}\end{array}$
Centralizer type: $C_4$
Unfold the hidden panel for more information.

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$A^{12}_1$
h-characteristic: (0, 1, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 24
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $A^{2}_2+4A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_2+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{2}_2+2A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{4\omega_{1}}+8V_{3\omega_{1}}+15V_{2\omega_{1}}+16V_{\omega_{1}}+12V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+16h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&g_{58}+g_{52}+2g_{49}+g_{44}+g_{34}+2g_{30}\\ f&=&g_{-30}+g_{-34}+g_{-44}+g_{-49}+g_{-52}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+16h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&2g_{58}+2g_{52}+4g_{49}+2g_{44}+2g_{34}+4g_{30}\\ [h, f]&=&-2g_{-30}-2g_{-34}-2g_{-44}-2g_{-49}-2g_{-52}-2g_{-58}\end{array}$
Centralizer type: $2A^{4}_1+A^{3}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 12): $g_{-8}$, $h_{7}-h_{1}$, $h_{8}$, $g_{1}+g_{-22}$, $g_{3}-g_{-3}$, $g_{4}-g_{-4}$, $g_{5}-g_{-5}$, $g_{8}$, $g_{11}-g_{-11}$, $g_{12}-g_{-12}$, $g_{18}-g_{-18}$, $g_{22}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{7}-h_{1}$, $-h_{8}$
Cartan of centralizer (dimension: 4): $g_{22}-g_{8}+g_{1}+h_{7}-h_{1}+g_{-1}-g_{-8}+g_{-22}$, $-2g_{22}-2g_{1}-h_{8}-2h_{7}+2h_{1}-2g_{-1}-2g_{-22}$, $g_{18}+g_{5}+g_{4}+g_{3}-g_{-3}-g_{-4}-g_{-5}-g_{-18}$, $g_{12}+g_{11}-g_{-11}-g_{-12}$
Cartan-generating semisimple element: $g_{22}+g_{18}+g_{12}+g_{11}+g_{8}+g_{5}+g_{4}+g_{3}+g_{1}+h_{8}+h_{7}-h_{1}+g_{-1}-g_{-3}-g_{-4}-g_{-5}+g_{-8}-g_{-11}-g_{-12}-g_{-18}+g_{-22}$
adjoint action: $\begin{pmatrix}-1 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1\\ 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & -1 & 1 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & -1 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & -1 & -1 & 0\\ 0 & 1 & -2 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 1 & -1 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1 & -1 & 1 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & -1 & 1 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{12}+2x^{10}-63x^8-20x^6+800x^4$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x^2-5)(x^2-5)(x^2+4)(x^2+8)
Eigenvalues of ad H: $0$, $\sqrt{5}$, $-\sqrt{5}$, $2\sqrt{-1}$, $-2\sqrt{-1}$, $2\sqrt{-2}$, $-2\sqrt{-2}$
12 eigenvectors of ad H: $(1,0,12,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,1,0,0)$, $(0,0,0,0,1,1,1,0,0,0,1,0)$, $(0,1,12,1,0,0,0,0,0,0,0,1)$, $(125−32,0,−125+12,0,0,0,0,1,0,0,0,0)$, $(0,−5+1,−125+12,125−32,0,0,0,0,0,0,0,1)$, $(−125−32,0,125+12,0,0,0,0,1,0,0,0,0)$, $(0,5+1,125+12,−125−32,0,0,0,0,0,0,0,1)$, $(0,0,0,0,−−1,−1,−1,0,0,0,1,0)$, $(0,0,0,0,−1,−1,−−1,0,0,0,1,0)$, $(0,0,0,0,−1,1,−1,0,−−2,−2,1,0)$, $(0,0,0,0,−1,1,−1,0,−2,−−2,1,0)$
Centralizer type: 2A^{4}_1+A^{3}_1+A^{1}_1
Reductive components (4 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$2/5\sqrt{5}g_{8}+1/5\sqrt{5}h_{8}+2/5\sqrt{5}g_{-8}$
matching e: $g_{8}+\left(-1/2\sqrt{5}+1/2\right)h_{8}+\left(1/2\sqrt{5}-3/2\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2/5\sqrt{5} & -2/5\sqrt{5} & 4/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5\sqrt{5} & -4/5\sqrt{5} & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2/5\sqrt{5}g_{8}+1/5\sqrt{5}h_{8}+2/5\sqrt{5}g_{-8}$
matching e: $g_{8}+\left(-1/2\sqrt{5}+1/2\right)h_{8}+\left(1/2\sqrt{5}-3/2\right)g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2/5\sqrt{5} & -2/5\sqrt{5} & 4/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5\sqrt{5} & -4/5\sqrt{5} & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$2/5\sqrt{5}g_{22}+2/5\sqrt{5}g_{1}+1/5\sqrt{5}h_{8}+2/5\sqrt{5}h_{7}-2/5\sqrt{5}h_{1}+2/5\sqrt{5}g_{-1}+2/5\sqrt{5}g_{-22}$
matching e: $g_{22}+\left(1/2\sqrt{5}-3/2\right)g_{1}+\left(-1/2\sqrt{5}+1/2\right)h_{8}+\left(-\sqrt{5}+1\right)h_{7}+\left(\sqrt{5}-1\right)h_{1}+g_{-1}+\left(1/2\sqrt{5}-3/2\right)g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 4/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4/5\sqrt{5}\\ 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5}\\ 0 & 2/5\sqrt{5} & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5}\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2/5\sqrt{5}g_{22}+2/5\sqrt{5}g_{1}+1/5\sqrt{5}h_{8}+2/5\sqrt{5}h_{7}-2/5\sqrt{5}h_{1}+2/5\sqrt{5}g_{-1}+2/5\sqrt{5}g_{-22}$
matching e: $g_{22}+\left(1/2\sqrt{5}-3/2\right)g_{1}+\left(-1/2\sqrt{5}+1/2\right)h_{8}+\left(-\sqrt{5}+1\right)h_{7}+\left(\sqrt{5}-1\right)h_{1}+g_{-1}+\left(1/2\sqrt{5}-3/2\right)g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 4/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4/5\sqrt{5}\\ 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5}\\ 0 & 2/5\sqrt{5} & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5}\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-\sqrt{-1}g_{12}-\sqrt{-1}g_{11}+\sqrt{-1}g_{-11}+\sqrt{-1}g_{-12}$
matching e: $g_{18}+\sqrt{-1}g_{5}-g_{4}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}+g_{-4}-\sqrt{-1}g_{-5}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & \sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-\sqrt{-1}g_{12}-\sqrt{-1}g_{11}+\sqrt{-1}g_{-11}+\sqrt{-1}g_{-12}$
matching e: $g_{18}+\sqrt{-1}g_{5}-g_{4}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}+g_{-4}-\sqrt{-1}g_{-5}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & \sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-1/2\sqrt{-2}g_{18}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{4}-1/2\sqrt{-2}g_{3}+1/2\sqrt{-2}g_{-3}+1/2\sqrt{-2}g_{-4}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-18}$
matching e: $g_{18}+\sqrt{-2}g_{12}-\sqrt{-2}g_{11}-g_{5}+g_{4}-g_{3}+g_{-3}-g_{-4}+g_{-5}+\sqrt{-2}g_{-11}-\sqrt{-2}g_{-12}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & 0 & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 0 & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-1/2\sqrt{-2}g_{18}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{4}-1/2\sqrt{-2}g_{3}+1/2\sqrt{-2}g_{-3}+1/2\sqrt{-2}g_{-4}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-18}$
matching e: $g_{18}+\sqrt{-2}g_{12}-\sqrt{-2}g_{11}-g_{5}+g_{4}-g_{3}+g_{-3}-g_{-4}+g_{-5}+\sqrt{-2}g_{-11}-\sqrt{-2}g_{-12}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & 0 & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 0 & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{11}_1$
h-characteristic: (2, 0, 1, 0, 0, 0, 0, 0)
Length of the weight dual to h: 22
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $C^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+V_{4\omega_{1}}+10V_{3\omega_{1}}+3V_{2\omega_{1}}+10V_{\omega_{1}}+55V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&5h_{8}+10h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&4g_{62}+g_{58}+3g_{1}\\ f&=&g_{-1}+g_{-58}+g_{-62}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&5h_{8}+10h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&8g_{62}+2g_{58}+6g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-58}-2g_{-62}\end{array}$
Centralizer type: $C_5$
Unfold the hidden panel for more information.

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$A^{11}_1$
h-characteristic: (0, 1, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 22
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{2}_2+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_2+3A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{4\omega_{1}}+6V_{3\omega_{1}}+15V_{2\omega_{1}}+18V_{\omega_{1}}+16V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&7h_{8}+14h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&g_{55}+2g_{53}-g_{44}-2g_{24}\\ f&=&-g_{-24}-g_{-44}+g_{-53}+g_{-55}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&7h_{8}+14h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&2g_{55}+4g_{53}-2g_{44}-4g_{24}\\ [h, f]&=&2g_{-24}+2g_{-44}-2g_{-53}-2g_{-55}\end{array}$
Centralizer type: $B_2+A^{8}_1+A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 16): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $h_{3}$, $h_{6}-h_{1}$, $h_{7}$, $h_{8}$, $g_{1}-g_{-34}$, $g_{4}+g_{-11}$, $g_{7}$, $g_{8}$, $g_{11}+g_{-4}$, $g_{15}$, $g_{22}$, $g_{34}-g_{-1}$
Basis of centralizer intersected with cartan (dimension: 4): $-h_{8}$, $-h_{7}$, $-h_{3}$, $h_{6}-h_{1}$
Cartan of centralizer (dimension: 4): $-h_{3}$, $h_{6}-h_{1}$, $-h_{7}$, $-h_{8}$
Cartan-generating semisimple element: $-h_{8}-9h_{7}-5h_{6}+7h_{3}+5h_{1}$
adjoint action: $\begin{pmatrix}4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 11/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7/2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -11/2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{16}-539/4x^{14}+110443/16x^{12}-10892377/64x^{10}+66405907/32x^8-713332249/64x^6+65367225/4x^4$
Factorization of characteristic polynomial of ad H: 1/64(x )(x )(x )(x )(x -7)(x -5)(x -4)(x +4)(x +5)(x +7)(2x -11)(2x -7)(2x -3)(2x +3)(2x +7)(2x +11)
Eigenvalues of ad H: $0$, $7$, $5$, $4$, $-4$, $-5$, $-7$, $11/2$, $7/2$, $3/2$, $-3/2$, $-7/2$, $-11/2$
16 eigenvectors of ad H: $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$
Centralizer type: B^{1}_2+A^{8}_1+A^{3}_1
Reductive components (3 total):
Scalar product computed: $\begin{pmatrix}1/9 & -1/18\\ -1/18 & 1/18\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$-h_{8}-2h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$2h_{8}+2h_{7}$
matching e: $g_{15}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}-2h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$2h_{8}+2h_{7}$
matching e: $g_{15}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 0), (-1, 0), (2, 1), (-2, -1), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36 & -36\\ -36 & 72\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{8}-2h_{7}-2h_{6}+2h_{1}$
matching e: $g_{1}-g_{-34}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}-2h_{7}-2h_{6}+2h_{1}$
matching e: $g_{1}-g_{-34}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$4h_{3}$
matching e: $g_{11}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$4h_{3}$
matching e: $g_{11}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}288\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{10}_1$
h-characteristic: (2, 1, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+12V_{3\omega_{1}}+V_{2\omega_{1}}+78V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&4h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+6h_{1}\\ e&=&4g_{62}+3g_{1}\\ f&=&g_{-1}+g_{-62}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&4h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&8g_{62}+6g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-62}\end{array}$
Centralizer type: $C_6$
Unfold the hidden panel for more information.

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$A^{10}_1$
h-characteristic: (0, 1, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{2}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_2+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{4\omega_{1}}+4V_{3\omega_{1}}+16V_{2\omega_{1}}+16V_{\omega_{1}}+25V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&6h_{8}+12h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&g_{58}+2g_{56}+g_{52}+2g_{17}\\ f&=&g_{-17}+g_{-52}+g_{-56}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&6h_{8}+12h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&2g_{58}+4g_{56}+2g_{52}+4g_{17}\\ [h, f]&=&-2g_{-17}-2g_{-52}-2g_{-56}-2g_{-58}\end{array}$
Centralizer type: $C_3+A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 25): $g_{-34}$, $g_{-28}$, $g_{-22}$, $g_{-21}$, $g_{-15}$, $g_{-14}$, $g_{-8}$, $g_{-7}$, $g_{-6}$, $h_{5}-h_{1}$, $h_{6}$, $h_{7}$, $h_{8}$, $g_{1}+g_{-44}$, $g_{3}-g_{-3}$, $g_{6}$, $g_{7}$, $g_{8}$, $g_{14}$, $g_{15}$, $g_{21}$, $g_{22}$, $g_{28}$, $g_{34}$, $g_{44}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 4): $h_{5}-h_{1}$, $-h_{8}$, $-h_{7}$, $-h_{6}$
Cartan of centralizer (dimension: 5): $-h_{7}$, $-h_{6}$, $h_{5}-h_{1}$, $-h_{8}$, $g_{3}-g_{-3}$
Cartan-generating semisimple element: $-9h_{8}+7h_{7}+5h_{6}+h_{5}-h_{1}$
adjoint action: $\begin{pmatrix}-4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 21/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 23/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -29/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 25 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -27/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 27/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -25 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 29/2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -23/2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -21/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{25}-1291x^{23}+4674543/8x^{21}-1997462351/16x^{19}+3441686272361/256x^{17}-11049787068411/16x^{15}+1786351718615063/128x^{13}-7476599047022629/64x^{11}+101274541816996881/256x^9-31564288362182949/64x^7+804526423430625/4x^5$
Factorization of characteristic polynomial of ad H: 1/256(x )(x )(x )(x )(x )(x -25)(x -4)(x -3)(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(x +3)(x +4)(x +25)(2x -29)(2x -27)(2x -23)(2x -21)(2x +21)(2x +23)(2x +27)(2x +29)
Eigenvalues of ad H: $0$, $25$, $4$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-4$, $-25$, $29/2$, $27/2$, $23/2$, $21/2$, $-21/2$, $-23/2$, $-27/2$, $-29/2$
25 eigenvectors of ad H: $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: C^{1}_3+A^{3}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/18 & -1/36 & -1/18\\ -1/36 & 1/18 & 0\\ -1/18 & 0 & 1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (3 total):
$2h_{6}$
matching e: $g_{6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{6}$
matching e: $g_{6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (1, 0, 1), (-1, 0, -1), (1, 0, 2), (-1, 0, -2), (0, 0, 1), (0, 0, -1), (1, 0, 0), (-1, 0, 0), (2, 1, 2), (-2, -1, -2), (1, 1, 2), (-1, -1, -2), (1, 1, 0), (-1, -1, 0), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}72 & -36 & -36\\ -36 & 72 & 0\\ -36 & 0 & 36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}+2h_{6}+2h_{5}-2h_{1}$
matching e: $g_{44}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}+2h_{6}+2h_{5}-2h_{1}$
matching e: $g_{44}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{9}_1$
h-characteristic: (0, 1, 1, 0, 0, 0, 0, 0)
Length of the weight dual to h: 18
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{2}_2+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{4\omega_{1}}+2V_{3\omega_{1}}+18V_{2\omega_{1}}+10V_{\omega_{1}}+39V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&5h_{8}+10h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&2g_{59}+g_{58}+2g_{10}\\ f&=&g_{-10}+g_{-58}+g_{-59}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&5h_{8}+10h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&4g_{59}+2g_{58}+4g_{10}\\ [h, f]&=&-2g_{-10}-2g_{-58}-2g_{-59}\end{array}$
Centralizer type: $C_4+A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 39): $g_{-44}$, $g_{-39}$, $g_{-34}$, $g_{-33}$, $g_{-28}$, $g_{-27}$, $g_{-22}$, $g_{-21}$, $g_{-20}$, $g_{-15}$, $g_{-14}$, $g_{-13}$, $g_{-8}$, $g_{-7}$, $g_{-6}$, $g_{-5}$, $h_{4}-h_{1}$, $h_{5}$, $h_{6}$, $h_{7}$, $h_{8}$, $g_{1}+g_{-52}$, $g_{5}$, $g_{6}$, $g_{7}$, $g_{8}$, $g_{13}$, $g_{14}$, $g_{15}$, $g_{20}$, $g_{21}$, $g_{22}$, $g_{27}$, $g_{28}$, $g_{33}$, $g_{34}$, $g_{39}$, $g_{44}$, $g_{52}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 5): $h_{4}-h_{1}$, $-h_{8}$, $-h_{7}$, $-h_{6}$, $-h_{5}$
Cartan of centralizer (dimension: 5): $-h_{8}$, $-h_{7}$, $-h_{6}$, $-h_{5}$, $h_{4}-h_{1}$
Cartan-generating semisimple element: $-9h_{8}+7h_{7}+5h_{6}-4h_{5}+h_{4}-h_{1}$
adjoint action: $\begin{pmatrix}5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -11/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 15 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 23/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -17 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 25 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -27/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 27/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -25 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 17 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -23/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -15 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 11/2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{39}-3677/2x^{37}+22237359/16x^{35}-4643592167/8x^{33}+38536465212415/256x^{31}-13162390357243905/512x^{29}+12219400883265204705/4096x^{27}-978486821568756140955/4096x^{25}+54303145496055821055465/4096x^{23}-2079438014416691460092215/4096x^{21}+54284680640906011787389399/4096x^{19}-947531534376194222383197249/4096x^{17}+10762144419009511371761880691/4096x^{15}-76737090176424233697408908341/4096x^{13}+81895838245603252374011075775/1024x^{11}-49128331796363050680576999375/256x^9+14714300153860535896903265625/64x^7-101654061370919789853515625x^5$
Factorization of characteristic polynomial of ad H: 1/4096(x )(x )(x )(x )(x )(x -25)(x -17)(x -15)(x -10)(x -9)(x -8)(x -7)(x -5)(x -2)(x -2)(x -1)(x +1)(x +2)(x +2)(x +5)(x +7)(x +8)(x +9)(x +10)(x +15)(x +17)(x +25)(2x -27)(2x -23)(2x -11)(2x -7)(2x -7)(2x -3)(2x +3)(2x +7)(2x +7)(2x +11)(2x +23)(2x +27)
Eigenvalues of ad H: $0$, $25$, $17$, $15$, $10$, $9$, $8$, $7$, $5$, $2$, $1$, $-1$, $-2$, $-5$, $-7$, $-8$, $-9$, $-10$, $-15$, $-17$, $-25$, $27/2$, $23/2$, $11/2$, $7/2$, $3/2$, $-3/2$, $-7/2$, $-11/2$, $-23/2$, $-27/2$
39 eigenvectors of ad H: $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: C^{1}_4+A^{3}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/18 & -1/36 & 0 & -1/36\\ -1/36 & 1/18 & -1/18 & 0\\ 0 & -1/18 & 1/9 & 0\\ -1/36 & 0 & 0 & 1/18\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (4 total):
$2h_{8}+4h_{7}+4h_{6}+2h_{5}$
matching e: $g_{39}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 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$-2h_{8}-4h_{7}-2h_{6}-2h_{5}$
matching e: $g_{-33}$
verification: $[h,e]-2e=0$
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$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
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$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{8}+4h_{7}+4h_{6}+2h_{5}$
matching e: $g_{39}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 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$-2h_{8}-4h_{7}-2h_{6}-2h_{5}$
matching e: $g_{-33}$
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$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
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$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1, 1), (-1, -1, -1, -1), (2, 2, 2, 1), (-2, -2, -2, -1), (1, 2, 2, 1), (-1, -2, -2, -1), (1, 0, 0, 1), (-1, 0, 0, -1), (1, 1, 1, 0), (-1, -1, -1, 0), (0, 0, 0, 1), (0, 0, 0, -1), (1, 2, 2, 0), (-1, -2, -2, 0), (0, 1, 1, 0), (0, -1, -1, 0), (1, 0, 0, 0), (-1, 0, 0, 0), (0, 0, 1, 0), (0, 0, -1, 0), (1, 1, 2, 1), (-1, -1, -2, -1), (1, 1, 0, 1), (-1, -1, 0, -1), (1, 1, 2, 0), (-1, -1, -2, 0), (0, 1, 2, 0), (0, -1, -2, 0), (1, 1, 0, 0), (-1, -1, 0, 0), (0, 1, 0, 0), (0, -1, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}72 & -36 & 0 & -36\\ -36 & 72 & -36 & 0\\ 0 & -36 & 36 & 0\\ -36 & 0 & 0 & 72\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}-2h_{1}$
matching e: $g_{52}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}-2h_{1}$
matching e: $g_{52}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

---

$A^{8}_1$
h-characteristic: (0, 0, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 16
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $4A^{2}_1$ Containing regular semisimple subalgebra number 2: $8A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{2}_1+6A^{1}_1$ Containing regular semisimple subalgebra number 4: $2A^{2}_1+4A^{1}_1$ Containing regular semisimple subalgebra number 5: $3A^{2}_1+2A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $36V_{2\omega_{1}}+28V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+14h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&g_{63}+g_{55}+g_{39}+g_{15}\\ f&=&g_{-15}+g_{-39}+g_{-55}+g_{-63}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+14h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{63}+2g_{55}+2g_{39}+2g_{15}\\ [h, f]&=&-2g_{-15}-2g_{-39}-2g_{-55}-2g_{-63}\end{array}$
Centralizer type: $D^{4}_4$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 28): $h_{1}$, $h_{3}$, $h_{5}$, $h_{7}$, $g_{2}-g_{-16}$, $g_{4}-g_{-18}$, $g_{6}-g_{-20}$, $g_{9}-g_{-10}$, $g_{10}-g_{-9}$, $g_{11}-g_{-12}$, $g_{12}-g_{-11}$, $g_{13}-g_{-14}$, $g_{14}-g_{-13}$, $g_{16}-g_{-2}$, $g_{17}-g_{-29}$, $g_{18}-g_{-4}$, $g_{19}-g_{-31}$, $g_{20}-g_{-6}$, $g_{23}-g_{-24}$, $g_{24}-g_{-23}$, $g_{25}-g_{-26}$, $g_{26}-g_{-25}$, $g_{29}-g_{-17}$, $g_{30}-g_{-40}$, $g_{31}-g_{-19}$, $g_{35}-g_{-36}$, $g_{36}-g_{-35}$, $g_{40}-g_{-30}$
Basis of centralizer intersected with cartan (dimension: 4): $-h_{7}$, $-h_{5}$, $-h_{3}$, $-h_{1}$
Cartan of centralizer (dimension: 4): $-h_{7}$, $-h_{5}$, $-h_{3}$, $-h_{1}$
Cartan-generating semisimple element: $-h_{7}-9h_{5}+7h_{3}+5h_{1}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{28}-234x^{26}+22815x^{24}-1218360x^{22}+39465855x^{20}-812168370x^{18}+10810706385x^{16}-92892784140x^{14}+504720730800x^{12}-1660725352000x^{10}+3065816991744x^8-2763360976896x^6+936404582400x^4$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -8)(x -7)(x -6)(x -5)(x -4)(x -4)(x -3)(x -3)(x -2)(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(x +2)(x +3)(x +3)(x +4)(x +4)(x +5)(x +6)(x +7)(x +8)
Eigenvalues of ad H: $0$, $8$, $7$, $6$, $5$, $4$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-4$, $-5$, $-6$, $-7$, $-8$
28 eigenvectors of ad H: $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: D^{4}_4
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/36 & -1/72 & 0 & 0\\ -1/72 & 1/36 & -1/72 & -1/72\\ 0 & -1/72 & 1/36 & 0\\ 0 & -1/72 & 0 & 1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (4 total):
$-2h_{7}+2h_{1}$
matching e: $g_{35}-g_{-36}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$2h_{3}-2h_{1}$
matching e: $g_{10}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
$2h_{7}+2h_{1}$
matching e: $g_{40}-g_{-30}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$-2h_{5}-2h_{3}$
matching e: $g_{4}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2h_{7}+2h_{1}$
matching e: $g_{35}-g_{-36}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$2h_{3}-2h_{1}$
matching e: $g_{10}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
$2h_{7}+2h_{1}$
matching e: $g_{40}-g_{-30}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$-2h_{5}-2h_{3}$
matching e: $g_{4}-g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 2, 1, 1), (-1, -2, -1, -1), (1, 1, 1, 1), (-1, -1, -1, -1), (1, 1, 1, 0), (-1, -1, -1, 0), (1, 1, 0, 1), (-1, -1, 0, -1), (0, 1, 1, 1), (0, -1, -1, -1), (1, 1, 0, 0), (-1, -1, 0, 0), (0, 1, 1, 0), (0, -1, -1, 0), (1, 0, 0, 0), (-1, 0, 0, 0), (0, 0, 1, 0), (0, 0, -1, 0), (0, 1, 0, 1), (0, -1, 0, -1), (0, 0, 0, 1), (0, 0, 0, -1), (0, 1, 0, 0), (0, -1, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144 & -72 & 0 & 0\\ -72 & 144 & -72 & -72\\ 0 & -72 & 144 & 0\\ 0 & -72 & 0 & 144\\ \end{pmatrix}$
Unfold the hidden panel for more information.

---

$A^{8}_1$
h-characteristic: (0, 2, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 16
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{2}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{4\omega_{1}}+21V_{2\omega_{1}}+58V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&4h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+4h_{1}\\ e&=&2g_{61}-2g_{2}\\ f&=&-g_{-2}+g_{-61}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&4h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&4g_{61}-4g_{2}\\ [h, f]&=&2g_{-2}-2g_{-61}\end{array}$
Centralizer type: $C_5+A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 58): $g_{-52}$, $g_{-48}$, $g_{-44}$, $g_{-43}$, $g_{-39}$, $g_{-38}$, $g_{-34}$, $g_{-33}$, $g_{-32}$, $g_{-28}$, $g_{-27}$, $g_{-26}$, $g_{-22}$, $g_{-21}$, $g_{-20}$, $g_{-19}$, $g_{-15}$, $g_{-14}$, $g_{-13}$, $g_{-12}$, $g_{-8}$, $g_{-7}$, $g_{-6}$, $g_{-5}$, $g_{-4}$, $h_{3}-h_{1}$, $h_{4}$, $h_{5}$, $h_{6}$, $h_{7}$, $h_{8}$, $g_{1}-g_{-58}$, $g_{4}$, $g_{5}$, $g_{6}$, $g_{7}$, $g_{8}$, $g_{12}$, $g_{13}$, $g_{14}$, $g_{15}$, $g_{19}$, $g_{20}$, $g_{21}$, $g_{22}$, $g_{26}$, $g_{27}$, $g_{28}$, $g_{32}$, $g_{33}$, $g_{34}$, $g_{38}$, $g_{39}$, $g_{43}$, $g_{44}$, $g_{48}$, $g_{52}$, $g_{58}-g_{-1}$
Basis of centralizer intersected with cartan (dimension: 6): $h_{3}-h_{1}$, $-h_{8}$, $-h_{7}$, $-h_{6}$, $-h_{5}$, $-h_{4}$
Cartan of centralizer (dimension: 6): $h_{3}-h_{1}$, $-h_{5}$, $-h_{4}$, $-h_{6}$, $-h_{8}$, $-h_{7}$
Cartan-generating semisimple element: $-9h_{8}+7h_{7}+5h_{6}-4h_{5}-3h_{4}+h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 5/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -5/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 27/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -25 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -13/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 17 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -23/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 21/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -13 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 11/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -29/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{58}-2182x^{56}+16463133/8x^{54}-18017073915/16x^{52}+103075704681495/256x^{50}-25630665156940725/256x^{48}+18383319452300976915/1024x^{46}-4859633577492748717635/2048x^{44}+15353649388039074533027775/65536x^{42}-142632881927825526879192985/8192x^{40}+512730729456799497143880904909/524288x^{38}-43605956446662255808487659262271/1048576x^{36}+22466282485814747899979550637305993/16777216x^{34}-547350875470840161006948674131941885/16777216x^{32}+1258768694958170826961247067032411835/2097152x^{30}-34864284835820272511722477327026338625/4194304x^{28}+723299911465923477474882350679887106495/8388608x^{26}-5580551982861606512294452547223414260715/8388608x^{24}+15859149681200441552046085124435780027275/4194304x^{22}-65560657739118342889721468373259785212905/4194304x^{20}+775477914130942905839300742314314267327521/16777216x^{18}-1605120076287567390763161931305598933995477/16777216x^{16}+564584993491366912355893368806730016534029/4194304x^{14}-32336086305635838515613212641940843317275/262144x^{12}+1125136960336109223629361275645807431875/16384x^{10}-20798911191258429380053625805865453125/1024x^8+36326669936817622716496730478515625/16x^6$
Factorization of characteristic polynomial of ad H: 1/16777216(x )(x )(x )(x )(x )(x )(x -25)(x -17)(x -13)(x -12)(x -9)(x -8)(x -5)(x -4)(x -4)(x -3)(x -2)(x -1)(x -1)(x -1)(x +1)(x +1)(x +1)(x +2)(x +3)(x +4)(x +4)(x +5)(x +8)(x +9)(x +12)(x +13)(x +17)(x +25)(2x -29)(2x -27)(2x -23)(2x -21)(2x -13)(2x -11)(2x -7)(2x -5)(2x -5)(2x -3)(2x -3)(2x -1)(2x +1)(2x +3)(2x +3)(2x +5)(2x +5)(2x +7)(2x +11)(2x +13)(2x +21)(2x +23)(2x +27)(2x +29)
Eigenvalues of ad H: $0$, $25$, $17$, $13$, $12$, $9$, $8$, $5$, $4$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-4$, $-5$, $-8$, $-9$, $-12$, $-13$, $-17$, $-25$, $29/2$, $27/2$, $23/2$, $21/2$, $13/2$, $11/2$, $7/2$, $5/2$, $3/2$, $1/2$, $-1/2$, $-3/2$, $-5/2$, $-7/2$, $-11/2$, $-13/2$, $-21/2$, $-23/2$, $-27/2$, $-29/2$
58 eigenvectors of ad H: $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$
Centralizer type: C^{1}_5+A^{3}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/18 & -1/36 & 0 & 0 & -1/36\\ -1/36 & 1/18 & -1/18 & 0 & 0\\ 0 & -1/18 & 1/9 & 0 & 0\\ 0 & 0 & 0 & 1/18 & -1/36\\ -1/36 & 0 & 0 & -1/36 & 1/18\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (5 total):
$-2h_{8}-4h_{7}-2h_{6}-2h_{5}-2h_{4}$
matching e: $g_{-38}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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$2h_{8}+4h_{7}+2h_{6}+2h_{5}$
matching e: $g_{33}$
verification: $[h,e]-2e=0$
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$-h_{8}-2h_{7}-2h_{6}-2h_{5}$
matching e: $g_{-44}$
verification: $[h,e]-2e=0$
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$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
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$2h_{8}+4h_{7}+4h_{6}+2h_{5}+2h_{4}$
matching e: $g_{43}$
verification: $[h,e]-2e=0$
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Linear space basis of intersection of centralizer and ambient Cartan:
$-2h_{8}-4h_{7}-2h_{6}-2h_{5}-2h_{4}$
matching e: $g_{-38}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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$2h_{8}+4h_{7}+2h_{6}+2h_{5}$
matching e: $g_{33}$
verification: $[h,e]-2e=0$
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$-h_{8}-2h_{7}-2h_{6}-2h_{5}$
matching e: $g_{-44}$
verification: $[h,e]-2e=0$
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$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
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$2h_{8}+4h_{7}+4h_{6}+2h_{5}+2h_{4}$
matching e: $g_{43}$
verification: $[h,e]-2e=0$
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Elements in Cartan dual to root system: (1, 1, 1, 1, 1), (-1, -1, -1, -1, -1), (2, 2, 2, 1, 2), (-2, -2, -2, -1, -2), (1, 1, 2, 1, 1), (-1, -1, -2, -1, -1), (1, 1, 0, 1, 1), (-1, -1, 0, -1, -1), (1, 1, 1, 0, 1), (-1, -1, -1, 0, -1), (0, 0, 0, 1, 0), (0, 0, 0, -1, 0), (1, 1, 2, 0, 1), (-1, -1, -2, 0, -1), (1, 1, 0, 0, 1), (-1, -1, 0, 0, -1), (0, 0, 1, 0, 0), (0, 0, -1, 0, 0), (1, 1, 1, 0, 0), (-1, -1, -1, 0, 0), (1, 2, 2, 0, 0), (-1, -2, -2, 0, 0), (0, 1, 1, 0, 0), (0, -1, -1, 0, 0), (1, 0, 0, 0, 0), (-1, 0, 0, 0, 0), (2, 2, 2, 1, 1), (-2, -2, -2, -1, -1), (1, 2, 2, 1, 1), (-1, -2, -2, -1, -1), (1, 0, 0, 1, 1), (-1, 0, 0, -1, -1), (0, 0, 0, 1, 1), (0, 0, 0, -1, -1), (2, 2, 2, 0, 1), (-2, -2, -2, 0, -1), (1, 2, 2, 0, 1), (-1, -2, -2, 0, -1), (1, 0, 0, 0, 1), (-1, 0, 0, 0, -1), (0, 0, 0, 0, 1), (0, 0, 0, 0, -1), (1, 1, 2, 0, 0), (-1, -1, -2, 0, 0), (1, 1, 0, 0, 0), (-1, -1, 0, 0, 0), (0, 1, 2, 0, 0), (0, -1, -2, 0, 0), (0, 1, 0, 0, 0), (0, -1, 0, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}72 & -36 & 0 & 0 & -36\\ -36 & 72 & -36 & 0 & 0\\ 0 & -36 & 36 & 0 & 0\\ 0 & 0 & 0 & 72 & -36\\ -36 & 0 & 0 & -36 & 72\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/27\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}-2h_{1}$
matching e: $g_{58}-g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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Linear space basis of intersection of centralizer and ambient Cartan:
$h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}-2h_{1}$
matching e: $g_{58}-g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}108\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{7}_1$
h-characteristic: (0, 0, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 14
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $3A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 2: $7A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{2}_1+5A^{1}_1$ Containing regular semisimple subalgebra number 4: $2A^{2}_1+3A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $28V_{2\omega_{1}}+14V_{\omega_{1}}+24V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&7h_{8}+14h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&g_{63}+g_{55}+g_{39}+g_{22}\\ f&=&g_{-22}+g_{-39}+g_{-55}+g_{-63}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&7h_{8}+14h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{63}+2g_{55}+2g_{39}+2g_{22}\\ [h, f]&=&-2g_{-22}-2g_{-39}-2g_{-55}-2g_{-63}\end{array}$
Centralizer type: $B^{4}_3+A_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 24): $g_{-8}$, $h_{1}$, $h_{3}$, $h_{5}$, $h_{8}$, $g_{2}-g_{-16}$, $g_{4}-g_{-18}$, $g_{6}-g_{-13}$, $g_{8}$, $g_{9}-g_{-10}$, $g_{10}-g_{-9}$, $g_{11}-g_{-12}$, $g_{12}-g_{-11}$, $g_{13}-g_{-6}$, $g_{16}-g_{-2}$, $g_{17}-g_{-29}$, $g_{18}-g_{-4}$, $g_{19}-g_{-25}$, $g_{23}-g_{-24}$, $g_{24}-g_{-23}$, $g_{25}-g_{-19}$, $g_{29}-g_{-17}$, $g_{30}-g_{-35}$, $g_{35}-g_{-30}$
Basis of centralizer intersected with cartan (dimension: 4): $-h_{1}$, $-h_{8}$, $-h_{5}$, $-h_{3}$
Cartan of centralizer (dimension: 4): $-h_{1}$, $-h_{8}$, $-h_{5}$, $-h_{3}$
Cartan-generating semisimple element: $-9h_{8}+7h_{5}+5h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5/2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{24}-1671/4x^{22}+540951/16x^{20}-75404621/64x^{18}+1402610379/64x^{16}-15083912571/64x^{14}+95250313549/64x^{12}-85049203419/16x^{10}+38423556819/4x^8-6721045389x^6+1157360400x^4$
Factorization of characteristic polynomial of ad H: 1/64(x )(x )(x )(x )(x -18)(x -6)(x -4)(x -3)(x -3)(x -2)(x -1)(x +1)(x +2)(x +3)(x +3)(x +4)(x +6)(x +18)(2x -7)(2x -5)(2x -1)(2x +1)(2x +5)(2x +7)
Eigenvalues of ad H: $0$, $18$, $6$, $4$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-4$, $-6$, $-18$, $7/2$, $5/2$, $1/2$, $-1/2$, $-5/2$, $-7/2$
24 eigenvectors of ad H: $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: B^{4}_3+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36 & -1/72 & -1/72\\ -1/72 & 1/72 & 0\\ -1/72 & 0 & 1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (3 total):
$2h_{3}+2h_{1}$
matching e: $g_{16}-g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
$-4h_{1}$
matching e: $g_{30}-g_{-35}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
$2h_{5}-2h_{3}$
matching e: $g_{12}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{3}+2h_{1}$
matching e: $g_{16}-g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
$-4h_{1}$
matching e: $g_{30}-g_{-35}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
$2h_{5}-2h_{3}$
matching e: $g_{12}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (2, 1, 1), (-2, -1, -1), (1, 1, 1), (-1, -1, -1), (1, 0, 1), (-1, 0, -1), (1, 1, 0), (-1, -1, 0), (1, 0, 0), (-1, 0, 0), (0, 0, 1), (0, 0, -1), (2, 1, 2), (-2, -1, -2), (2, 1, 0), (-2, -1, 0), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144 & -144 & -72\\ -144 & 288 & 0\\ -72 & 0 & 144\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{6}_1$
h-characteristic: (0, 0, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 12
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $3A^{2}_1$ Containing regular semisimple subalgebra number 2: $6A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{2}_1+4A^{1}_1$ Containing regular semisimple subalgebra number 4: $2A^{2}_1+2A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $21V_{2\omega_{1}}+24V_{\omega_{1}}+25V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&6h_{8}+12h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&g_{63}-g_{55}+g_{39}\\ f&=&g_{-39}-g_{-55}+g_{-63}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&6h_{8}+12h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{63}-2g_{55}+2g_{39}\\ [h, f]&=&-2g_{-39}+2g_{-55}-2g_{-63}\end{array}$
Centralizer type: $A^{4}_3+B_2$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 25): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $h_{1}$, $h_{3}$, $h_{5}$, $h_{7}$, $h_{8}$, $g_{2}+g_{-16}$, $g_{4}+g_{-18}$, $g_{7}$, $g_{8}$, $g_{9}+g_{-10}$, $g_{10}+g_{-9}$, $g_{11}+g_{-12}$, $g_{12}+g_{-11}$, $g_{15}$, $g_{16}+g_{-2}$, $g_{17}-g_{-29}$, $g_{18}+g_{-4}$, $g_{22}$, $g_{23}-g_{-24}$, $g_{24}-g_{-23}$, $g_{29}-g_{-17}$
Basis of centralizer intersected with cartan (dimension: 5): $-h_{7}$, $-h_{5}$, $-h_{3}$, $-h_{1}$, $-h_{8}$
Cartan of centralizer (dimension: 5): $-g_{22}-g_{15}-g_{8}-3h_{8}-3h_{7}-g_{-8}-g_{-15}-g_{-22}$, $-1/2g_{29}-1/2g_{23}-1/2g_{18}-1/2g_{17}-1/2g_{16}-1/2g_{11}-1/2g_{9}-1/2g_{4}-1/2g_{2}-h_{3}-1/2g_{-2}-1/2g_{-4}-1/2g_{-10}-1/2g_{-12}-1/2g_{-16}+1/2g_{-17}-1/2g_{-18}+1/2g_{-24}+1/2g_{-29}$, $-g_{24}-g_{23}-g_{12}-g_{11}-g_{10}-g_{9}-g_{-9}-g_{-10}-g_{-11}-g_{-12}+g_{-23}+g_{-24}$, $-g_{7}+2h_{8}+2h_{7}-g_{-7}$, $-1/2g_{29}+1/2g_{23}-1/2g_{18}-1/2g_{17}-1/2g_{16}+1/2g_{11}+1/2g_{9}-1/2g_{4}-1/2g_{2}-h_{5}-h_{1}-1/2g_{-2}-1/2g_{-4}+1/2g_{-10}+1/2g_{-12}-1/2g_{-16}+1/2g_{-17}-1/2g_{-18}-1/2g_{-24}+1/2g_{-29}$
Cartan-generating semisimple element: $g_{29}+g_{24}+g_{23}+g_{22}+g_{18}+g_{17}+g_{16}+g_{15}+g_{12}+g_{11}+g_{10}+g_{9}+g_{8}+g_{7}+g_{4}+g_{2}+h_{8}+h_{7}+h_{5}+h_{3}+h_{1}+g_{-2}+g_{-4}+g_{-7}+g_{-8}+g_{-9}+g_{-10}+g_{-11}+g_{-12}+g_{-15}+g_{-16}-g_{-17}+g_{-18}+g_{-22}-g_{-23}-g_{-24}-g_{-29}$
adjoint action: $\begin{pmatrix}-1 & -2 & 0 & 2 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & -1 & -1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & -1 & 0 & 0 & 0 & 0 & -1 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & -2 & 2 & 0 & 0 & 0 & -2 & -2 & 0 & 0 & 2 & -2 & 2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 2 & -2 & -2 & 2 & 0 & -2 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & -2 & 0 & 0 & -2 & -2 & 0 & -2 & 2 & 2\\ 2 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/2 & 1/2 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1/2 & 1/2 & 0 & 0 & -1 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0\\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 1 & -2 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 1/2 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1\\ 0 & 0 & 0 & 0 & 1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1/2 & -1/2 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & -1 & 0 & -1\\ 0 & 0 & 0 & 0 & 1/2 & 0 & 1/2 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 0 & 1/2 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/2 & 0 & -1/2 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 0 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & -1 & 0 & 1 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{25}-58x^{23}+1273x^{21}-14446x^{19}+94740x^{17}-372894x^{15}+873557x^{13}-1154282x^{11}+767429x^9-213320x^7+18000x^5$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x )(x -5)(x -3)(x -2)(x -2)(x -1)(x +1)(x +2)(x +2)(x +3)(x +5)(x^2-5)(x^2-3x +1)(x^2-x -1)(x^2+x -1)(x^2+3x +1)
Eigenvalues of ad H: $0$, $5$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-5$, $\sqrt{5}$, $-\sqrt{5}$, $1/2\sqrt{5}+3/2$, $-1/2\sqrt{5}+3/2$, $1/2\sqrt{5}+1/2$, $-1/2\sqrt{5}+1/2$, $1/2\sqrt{5}-1/2$, $-1/2\sqrt{5}-1/2$, $1/2\sqrt{5}-3/2$, $-1/2\sqrt{5}-3/2$
25 eigenvectors of ad H: $(0,0,0,1,0,0,0,−2,−2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(1,1,1,0,0,0,0,3,3,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0)$, $(0,0,0,0,−1,1,−1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,1,−1,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0)$, $(0,0,0,0,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0,1)$, $(−14,−14,−14,−12,0,0,0,−1,−1,0,0,−12,1,0,0,0,0,1,0,0,0,1,0,0,0)$, $(−1,0,1,−1,0,0,0,−2,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,2,−2,0,0,0,0,0,0,0,−1,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,14,0,0,0,−12,0,0,0,−14,−1,0,0,0,0,0,0,0,0,1,0,0,0)$, $(1,−1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,−1,0,0,0,1,0,0,0)$, $(4,0,−4,−1,0,0,0,−2,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,−2,2,0,0,0,0,0,0,0,0,−1,1,0,0,0,0,0,0,0,0)$, $(0,0,0,−1,0,0,0,2,0,0,0,1,−1,0,0,0,0,0,0,0,0,1,0,0,0)$, $(−4,−4,−4,2,0,0,0,4,4,0,0,2,1,0,0,0,0,1,0,0,0,1,0,0,0)$, $(0,0,0,0,0,−25+2,0,0,0,125−32,125−32,0,0,−125+12,125−12,−125+12,125−12,0,1,125−32,1,0,−125+12,125−12,1)$, $(0,0,0,0,0,25+2,0,0,0,−125−32,−125−32,0,0,125+12,−125−12,125+12,−125−12,0,1,−125−32,1,0,125+12,−125−12,1)$, $(0,0,0,0,5−1,−5+1,0,0,0,0,0,0,0,−125+12,125−12,0,125−12,0,0,0,1,0,0,125−12,1)$, $(0,0,0,0,−5−1,5+1,0,0,0,0,0,0,0,125+12,−125−12,0,−125−12,0,0,0,1,0,0,−125−12,1)$, $(0,0,0,0,−2,2,0,0,0,0,125−12,0,0,1,−1,1,0,0,0,125−12,0,0,1,0,0)$, $(0,0,0,0,−2,2,0,0,0,0,−125−12,0,0,1,−1,1,0,0,0,−125−12,0,0,1,0,0)$, $(0,0,0,0,0,−5+1,5−1,0,0,0,0,0,0,−125+12,0,−125+12,125−12,0,1,0,0,0,−125+12,0,1)$, $(0,0,0,0,0,5+1,−5−1,0,0,0,0,0,0,125+12,0,125+12,−125−12,0,1,0,0,0,125+12,0,1)$, $(0,0,0,0,0,−2,2,0,0,−125+12,0,0,0,0,1,−1,1,0,0,−125+12,0,0,0,1,0)$, $(0,0,0,0,0,−2,2,0,0,125+12,0,0,0,0,1,−1,1,0,0,125+12,0,0,0,1,0)$
Centralizer type: A^{4}_3+B^{1}_2
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/18 & -1/18\\ -1/18 & 1/9\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$2/5g_{22}+2/5g_{15}+2/5g_{8}-1/5g_{7}+8/5h_{8}+8/5h_{7}-1/5g_{-7}+2/5g_{-8}+2/5g_{-15}+2/5g_{-22}$
matching e: $g_{22}-g_{8}-1/4g_{7}-1/2h_{7}+1/4g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-8/5 & 2/5 & 0 & 4/5 & 0 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/5 & -8/5 & 1/5 & 2/5 & 0 & 0 & 0 & 0 & 2/5 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & -8/5 & 0 & 0 & 0 & 0 & -2/5 & 4/5 & 0 & 0 & 4/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & -1/5 & 1/5 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 4/5 & 4/5 & 0 & -2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0 & 0 & -4/5 & 0 & 0 & 0 & -4/5 & 0 & 0 & 0\\ 2/5 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 0 & -4/5 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & 2/5 & 0 & 0 & 0 & 0 & 1/5 & -1/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0\\ 0 & 0 & 0 & -4/5 & 0 & 0 & 0 & 2/5 & -4/5 & 0 & 0 & 0 & 8/5 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & -2/5 & -1/5 & 0 & 0 & 0 & 0 & 8/5 & 0 & 0 & 0 & -1/5 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & -4/5 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 8/5 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$1/2g_{7}-h_{8}-h_{7}+1/2g_{-7}$
matching e: $g_{-8}-g_{-15}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2/5g_{22}+2/5g_{15}+2/5g_{8}-1/5g_{7}+8/5h_{8}+8/5h_{7}-1/5g_{-7}+2/5g_{-8}+2/5g_{-15}+2/5g_{-22}$
matching e: $g_{22}-g_{8}-1/4g_{7}-1/2h_{7}+1/4g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-8/5 & 2/5 & 0 & 4/5 & 0 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/5 & -8/5 & 1/5 & 2/5 & 0 & 0 & 0 & 0 & 2/5 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & -8/5 & 0 & 0 & 0 & 0 & -2/5 & 4/5 & 0 & 0 & 4/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & -1/5 & 1/5 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 4/5 & 4/5 & 0 & -2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0 & 0 & -4/5 & 0 & 0 & 0 & -4/5 & 0 & 0 & 0\\ 2/5 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 0 & -4/5 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & 2/5 & 0 & 0 & 0 & 0 & 1/5 & -1/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0\\ 0 & 0 & 0 & -4/5 & 0 & 0 & 0 & 2/5 & -4/5 & 0 & 0 & 0 & 8/5 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & -2/5 & -1/5 & 0 & 0 & 0 & 0 & 8/5 & 0 & 0 & 0 & -1/5 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & -4/5 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & 8/5 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$1/2g_{7}-h_{8}-h_{7}+1/2g_{-7}$
matching e: $g_{-8}-g_{-15}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}72 & -36\\ -36 & 36\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36 & 0 & -1/72\\ 0 & 1/36 & -1/72\\ -1/72 & -1/72 & 1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (3 total):
$-1/5\sqrt{5}g_{29}+\left(1/10\sqrt{5}+3/2\right)g_{24}+\left(-1/10\sqrt{5}-1/2\right)g_{23}-1/5\sqrt{5}g_{18}-1/5\sqrt{5}g_{17}-1/5\sqrt{5}g_{16}+\left(1/10\sqrt{5}+3/2\right)g_{12}+\left(-1/10\sqrt{5}-1/2\right)g_{11}+\left(1/10\sqrt{5}+3/2\right)g_{10}+\left(-1/10\sqrt{5}-1/2\right)g_{9}-1/5\sqrt{5}g_{4}-1/5\sqrt{5}g_{2}+2h_{5}+\left(-2/5\sqrt{5}-2\right)h_{3}+2h_{1}-1/5\sqrt{5}g_{-2}-1/5\sqrt{5}g_{-4}+\left(1/10\sqrt{5}+3/2\right)g_{-9}+\left(-1/10\sqrt{5}-1/2\right)g_{-10}+\left(1/10\sqrt{5}+3/2\right)g_{-11}+\left(-1/10\sqrt{5}-1/2\right)g_{-12}-1/5\sqrt{5}g_{-16}+1/5\sqrt{5}g_{-17}-1/5\sqrt{5}g_{-18}+\left(-1/10\sqrt{5}-3/2\right)g_{-23}+\left(1/10\sqrt{5}+1/2\right)g_{-24}+1/5\sqrt{5}g_{-29}$
matching e: $g_{29}+\left(-1/2\sqrt{5}-1/2\right)g_{24}+g_{18}+\left(-1/2\sqrt{5}-1/2\right)g_{12}+\left(-1/2\sqrt{5}-1/2\right)g_{10}+\left(1/2\sqrt{5}+1/2\right)g_{9}+\left(\sqrt{5}+1\right)h_{3}+\left(-\sqrt{5}-1\right)h_{1}+g_{-4}+\left(-1/2\sqrt{5}-1/2\right)g_{-9}+\left(1/2\sqrt{5}+1/2\right)g_{-10}+\left(-1/2\sqrt{5}-1/2\right)g_{-11}-g_{-17}+\left(1/2\sqrt{5}+1/2\right)g_{-23}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & -1/5\sqrt{5}-3 & -1/5\sqrt{5}-1 & 0 & 0 & 0 & 2/5\sqrt{5} & 2/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5}+3 & 1/5\sqrt{5}+1 & -2/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5} & -2/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5}+3 & 1/5\sqrt{5}+1 & -1/5\sqrt{5}-3 & -1/5\sqrt{5}-1 & 0 & 2/5\sqrt{5} & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 1/5\sqrt{5}+3 & 1/5\sqrt{5}+1 & 0 & 0 & 2/5\sqrt{5} & 2/5\sqrt{5} & 0 & -1/5\sqrt{5}-3 & -1/5\sqrt{5}-1 & -2/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/10\sqrt{5} & -1/10\sqrt{5} & 0 & 0 & 0 & 1/5\sqrt{5} & 1/10\sqrt{5}+3/2 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}-3/2 & 0 & 0 & 0 & 1/5\sqrt{5} & 0\\ 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5} & -1/10\sqrt{5} & 0 & 0 & 1/10\sqrt{5}+1/2 & 1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5}-1/2 & 0 & 0 & -1/5\sqrt{5} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/20\sqrt{5}+1/4 & -1/20\sqrt{5}-1/4 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5}+2 & 0 & 0 & -1/10\sqrt{5}-1/2 & 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5}-3/2 & 0 & 1/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 1/20\sqrt{5}+3/4 & -1/20\sqrt{5}-3/4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5}-2 & 1/10\sqrt{5}+3/2 & 0 & 0 & 0 & 1/5\sqrt{5} & -1/5\sqrt{5} & 0 & 0 & 1/10\sqrt{5}+1/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 1/20\sqrt{5}+1/4 & -1/20\sqrt{5}-1/4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}+1/2 & -1/5\sqrt{5}-2 & 0 & 0 & 1/5\sqrt{5} & -1/5\sqrt{5} & 0 & 0 & 1/10\sqrt{5}+3/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1/20\sqrt{5}+3/4 & -1/20\sqrt{5}-3/4 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & -1/10\sqrt{5}-3/2 & 0 & 0 & 1/5\sqrt{5}+2 & 0 & 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5}-1/2 & -1/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5} & 0 & -1/10\sqrt{5}-1/2 & 0 & 1/5\sqrt{5} & 0 & 1/10\sqrt{5}+1/2\\ 0 & 0 & 0 & 0 & -1/10\sqrt{5} & 0 & -1/10\sqrt{5} & 0 & 0 & 1/10\sqrt{5}+1/2 & 1/10\sqrt{5}+3/2 & 0 & 0 & 0 & 1/5\sqrt{5} & -1/5\sqrt{5} & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & -1/10\sqrt{5}-3/2 & 0 & -1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5} & 1/10\sqrt{5}+3/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/20\sqrt{5}+1/4 & 0 & -1/20\sqrt{5}-1/4 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 1/10\sqrt{5}+1/2 & 0 & -1/10\sqrt{5}-1/2 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/20\sqrt{5}+3/4 & 0 & -1/20\sqrt{5}-3/4 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & 0 & -1/10\sqrt{5}-3/2 & 0 & 1/10\sqrt{5}+3/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 0 & 1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5} & 0 & -1/10\sqrt{5}-3/2 & 0 & -1/10\sqrt{5}-1/2 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$1/5\sqrt{5}g_{29}+\left(-1/10\sqrt{5}+1/2\right)g_{24}+\left(1/10\sqrt{5}-3/2\right)g_{23}+1/5\sqrt{5}g_{18}+1/5\sqrt{5}g_{17}+1/5\sqrt{5}g_{16}+\left(-1/10\sqrt{5}+1/2\right)g_{12}+\left(1/10\sqrt{5}-3/2\right)g_{11}+\left(-1/10\sqrt{5}+1/2\right)g_{10}+\left(1/10\sqrt{5}-3/2\right)g_{9}+1/5\sqrt{5}g_{4}+1/5\sqrt{5}g_{2}+2h_{5}+\left(2/5\sqrt{5}-2\right)h_{3}+2h_{1}+1/5\sqrt{5}g_{-2}+1/5\sqrt{5}g_{-4}+\left(-1/10\sqrt{5}+1/2\right)g_{-9}+\left(1/10\sqrt{5}-3/2\right)g_{-10}+\left(-1/10\sqrt{5}+1/2\right)g_{-11}+\left(1/10\sqrt{5}-3/2\right)g_{-12}+1/5\sqrt{5}g_{-16}-1/5\sqrt{5}g_{-17}+1/5\sqrt{5}g_{-18}+\left(1/10\sqrt{5}-1/2\right)g_{-23}+\left(-1/10\sqrt{5}+3/2\right)g_{-24}-1/5\sqrt{5}g_{-29}$
matching e: $g_{29}+\left(-1/2\sqrt{5}+1/2\right)g_{23}+g_{16}+\left(1/2\sqrt{5}-1/2\right)g_{12}+\left(-1/2\sqrt{5}+1/2\right)g_{11}+\left(-1/2\sqrt{5}+1/2\right)g_{9}+\left(\sqrt{5}-1\right)h_{5}+\left(-\sqrt{5}+1\right)h_{3}+g_{-2}+\left(-1/2\sqrt{5}+1/2\right)g_{-10}+\left(1/2\sqrt{5}-1/2\right)g_{-11}+\left(-1/2\sqrt{5}+1/2\right)g_{-12}-g_{-17}+\left(1/2\sqrt{5}-1/2\right)g_{-24}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 1/5\sqrt{5}-1 & 1/5\sqrt{5}-3 & 0 & 0 & 0 & -2/5\sqrt{5} & -2/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5}+1 & -1/5\sqrt{5}+3 & 2/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 2/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5}+1 & -1/5\sqrt{5}+3 & 1/5\sqrt{5}-1 & 1/5\sqrt{5}-3 & 0 & -2/5\sqrt{5} & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0 & -1/5\sqrt{5}+1 & -1/5\sqrt{5}+3 & 0 & 0 & -2/5\sqrt{5} & -2/5\sqrt{5} & 0 & 1/5\sqrt{5}-1 & 1/5\sqrt{5}-3 & 2/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 1/10\sqrt{5} & 0 & 0 & 0 & -1/5\sqrt{5} & -1/10\sqrt{5}+1/2 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 1/10\sqrt{5}-1/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 0\\ 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 1/10\sqrt{5} & 0 & 0 & -1/10\sqrt{5}+3/2 & -1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}-3/2 & 0 & 0 & 1/5\sqrt{5} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}+3/4 & 1/20\sqrt{5}-3/4 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5}+2 & 0 & 0 & 1/10\sqrt{5}-3/2 & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}-1/2 & 0 & -1/5\sqrt{5}\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}+1/4 & 1/20\sqrt{5}-1/4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5}-2 & -1/10\sqrt{5}+1/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}+3/2 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/20\sqrt{5}+3/4 & 1/20\sqrt{5}-3/4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5}+3/2 & 1/5\sqrt{5}-2 & 0 & 0 & -1/5\sqrt{5} & 1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}+1/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/20\sqrt{5}+1/4 & 1/20\sqrt{5}-1/4 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 1/10\sqrt{5}-1/2 & 0 & 0 & -1/5\sqrt{5}+2 & 0 & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}-3/2 & 1/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/10\sqrt{5} & -1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & 0 & 1/10\sqrt{5}-3/2 & 0 & -1/5\sqrt{5} & 0 & -1/10\sqrt{5}+3/2\\ 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 0 & 1/10\sqrt{5} & 0 & 0 & -1/10\sqrt{5}+3/2 & -1/10\sqrt{5}+1/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 1/5\sqrt{5} & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5} & -1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 1/10\sqrt{5}-1/2 & 0 & 1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & -1/10\sqrt{5}+1/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}+3/4 & 0 & 1/20\sqrt{5}-3/4 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}+3/2 & 0 & 1/10\sqrt{5}-3/2 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}+1/4 & 0 & 1/20\sqrt{5}-1/4 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0 & 1/10\sqrt{5}-1/2 & 0 & -1/10\sqrt{5}+1/2 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/10\sqrt{5} & 0 & -1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & 0 & 1/10\sqrt{5}-1/2 & 0 & 1/10\sqrt{5}-3/2 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$1/5\sqrt{5}g_{29}+\left(-1/10\sqrt{5}-1/2\right)g_{24}+\left(1/10\sqrt{5}+3/2\right)g_{23}+1/5\sqrt{5}g_{18}+1/5\sqrt{5}g_{17}+1/5\sqrt{5}g_{16}+\left(-1/10\sqrt{5}-1/2\right)g_{12}+\left(1/10\sqrt{5}+3/2\right)g_{11}+\left(-1/10\sqrt{5}-1/2\right)g_{10}+\left(1/10\sqrt{5}+3/2\right)g_{9}+1/5\sqrt{5}g_{4}+1/5\sqrt{5}g_{2}-2h_{5}+\left(2/5\sqrt{5}+2\right)h_{3}-2h_{1}+1/5\sqrt{5}g_{-2}+1/5\sqrt{5}g_{-4}+\left(-1/10\sqrt{5}-1/2\right)g_{-9}+\left(1/10\sqrt{5}+3/2\right)g_{-10}+\left(-1/10\sqrt{5}-1/2\right)g_{-11}+\left(1/10\sqrt{5}+3/2\right)g_{-12}+1/5\sqrt{5}g_{-16}-1/5\sqrt{5}g_{-17}+1/5\sqrt{5}g_{-18}+\left(1/10\sqrt{5}+1/2\right)g_{-23}+\left(-1/10\sqrt{5}-3/2\right)g_{-24}-1/5\sqrt{5}g_{-29}$
matching e: $g_{23}+\left(1/2\sqrt{5}-1/2\right)g_{17}+g_{11}-g_{10}+g_{9}+\left(1/2\sqrt{5}-1/2\right)g_{4}+2h_{3}-2h_{1}-g_{-9}+g_{-10}+g_{-12}+\left(1/2\sqrt{5}-1/2\right)g_{-18}-g_{-24}+\left(-1/2\sqrt{5}+1/2\right)g_{-29}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 1/5\sqrt{5}+1 & 1/5\sqrt{5}+3 & 0 & 0 & 0 & -2/5\sqrt{5} & -2/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5}-1 & -1/5\sqrt{5}-3 & 2/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 2/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5}-1 & -1/5\sqrt{5}-3 & 1/5\sqrt{5}+1 & 1/5\sqrt{5}+3 & 0 & -2/5\sqrt{5} & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0 & -1/5\sqrt{5}-1 & -1/5\sqrt{5}-3 & 0 & 0 & -2/5\sqrt{5} & -2/5\sqrt{5} & 0 & 1/5\sqrt{5}+1 & 1/5\sqrt{5}+3 & 2/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 1/10\sqrt{5} & 0 & 0 & 0 & -1/5\sqrt{5} & -1/10\sqrt{5}-1/2 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 1/10\sqrt{5}+1/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 0\\ 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 1/10\sqrt{5} & 0 & 0 & -1/10\sqrt{5}-3/2 & -1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}+3/2 & 0 & 0 & 1/5\sqrt{5} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}-3/4 & 1/20\sqrt{5}+3/4 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5}-2 & 0 & 0 & 1/10\sqrt{5}+3/2 & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}+1/2 & 0 & -1/5\sqrt{5}\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}-1/4 & 1/20\sqrt{5}+1/4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5}+2 & -1/10\sqrt{5}-1/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}-3/2 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/20\sqrt{5}-3/4 & 1/20\sqrt{5}+3/4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5}-3/2 & 1/5\sqrt{5}+2 & 0 & 0 & -1/5\sqrt{5} & 1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}-1/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/20\sqrt{5}-1/4 & 1/20\sqrt{5}+1/4 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 1/10\sqrt{5}+1/2 & 0 & 0 & -1/5\sqrt{5}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}+3/2 & 1/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/10\sqrt{5} & -1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & 0 & 1/10\sqrt{5}+3/2 & 0 & -1/5\sqrt{5} & 0 & -1/10\sqrt{5}-3/2\\ 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 0 & 1/10\sqrt{5} & 0 & 0 & -1/10\sqrt{5}-3/2 & -1/10\sqrt{5}-1/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 1/5\sqrt{5} & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5} & -1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 1/10\sqrt{5}+1/2 & 0 & 1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & -1/10\sqrt{5}-1/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}-3/4 & 0 & 1/20\sqrt{5}+3/4 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}-3/2 & 0 & 1/10\sqrt{5}+3/2 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}-1/4 & 0 & 1/20\sqrt{5}+1/4 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0 & 1/10\sqrt{5}+1/2 & 0 & -1/10\sqrt{5}-1/2 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/10\sqrt{5} & 0 & -1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & 0 & 1/10\sqrt{5}+1/2 & 0 & 1/10\sqrt{5}+3/2 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-1/5\sqrt{5}g_{29}+\left(1/10\sqrt{5}+3/2\right)g_{24}+\left(-1/10\sqrt{5}-1/2\right)g_{23}-1/5\sqrt{5}g_{18}-1/5\sqrt{5}g_{17}-1/5\sqrt{5}g_{16}+\left(1/10\sqrt{5}+3/2\right)g_{12}+\left(-1/10\sqrt{5}-1/2\right)g_{11}+\left(1/10\sqrt{5}+3/2\right)g_{10}+\left(-1/10\sqrt{5}-1/2\right)g_{9}-1/5\sqrt{5}g_{4}-1/5\sqrt{5}g_{2}+2h_{5}+\left(-2/5\sqrt{5}-2\right)h_{3}+2h_{1}-1/5\sqrt{5}g_{-2}-1/5\sqrt{5}g_{-4}+\left(1/10\sqrt{5}+3/2\right)g_{-9}+\left(-1/10\sqrt{5}-1/2\right)g_{-10}+\left(1/10\sqrt{5}+3/2\right)g_{-11}+\left(-1/10\sqrt{5}-1/2\right)g_{-12}-1/5\sqrt{5}g_{-16}+1/5\sqrt{5}g_{-17}-1/5\sqrt{5}g_{-18}+\left(-1/10\sqrt{5}-3/2\right)g_{-23}+\left(1/10\sqrt{5}+1/2\right)g_{-24}+1/5\sqrt{5}g_{-29}$
matching e: $g_{29}+\left(-1/2\sqrt{5}-1/2\right)g_{24}+g_{18}+\left(-1/2\sqrt{5}-1/2\right)g_{12}+\left(-1/2\sqrt{5}-1/2\right)g_{10}+\left(1/2\sqrt{5}+1/2\right)g_{9}+\left(\sqrt{5}+1\right)h_{3}+\left(-\sqrt{5}-1\right)h_{1}+g_{-4}+\left(-1/2\sqrt{5}-1/2\right)g_{-9}+\left(1/2\sqrt{5}+1/2\right)g_{-10}+\left(-1/2\sqrt{5}-1/2\right)g_{-11}-g_{-17}+\left(1/2\sqrt{5}+1/2\right)g_{-23}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & -1/5\sqrt{5}-3 & -1/5\sqrt{5}-1 & 0 & 0 & 0 & 2/5\sqrt{5} & 2/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5}+3 & 1/5\sqrt{5}+1 & -2/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5} & -2/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5}+3 & 1/5\sqrt{5}+1 & -1/5\sqrt{5}-3 & -1/5\sqrt{5}-1 & 0 & 2/5\sqrt{5} & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 1/5\sqrt{5}+3 & 1/5\sqrt{5}+1 & 0 & 0 & 2/5\sqrt{5} & 2/5\sqrt{5} & 0 & -1/5\sqrt{5}-3 & -1/5\sqrt{5}-1 & -2/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/10\sqrt{5} & -1/10\sqrt{5} & 0 & 0 & 0 & 1/5\sqrt{5} & 1/10\sqrt{5}+3/2 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}-3/2 & 0 & 0 & 0 & 1/5\sqrt{5} & 0\\ 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5} & -1/10\sqrt{5} & 0 & 0 & 1/10\sqrt{5}+1/2 & 1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5}-1/2 & 0 & 0 & -1/5\sqrt{5} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/20\sqrt{5}+1/4 & -1/20\sqrt{5}-1/4 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5}+2 & 0 & 0 & -1/10\sqrt{5}-1/2 & 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5}-3/2 & 0 & 1/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 1/20\sqrt{5}+3/4 & -1/20\sqrt{5}-3/4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5}-2 & 1/10\sqrt{5}+3/2 & 0 & 0 & 0 & 1/5\sqrt{5} & -1/5\sqrt{5} & 0 & 0 & 1/10\sqrt{5}+1/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 1/20\sqrt{5}+1/4 & -1/20\sqrt{5}-1/4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}+1/2 & -1/5\sqrt{5}-2 & 0 & 0 & 1/5\sqrt{5} & -1/5\sqrt{5} & 0 & 0 & 1/10\sqrt{5}+3/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1/20\sqrt{5}+3/4 & -1/20\sqrt{5}-3/4 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & -1/10\sqrt{5}-3/2 & 0 & 0 & 1/5\sqrt{5}+2 & 0 & 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5}-1/2 & -1/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5} & 0 & -1/10\sqrt{5}-1/2 & 0 & 1/5\sqrt{5} & 0 & 1/10\sqrt{5}+1/2\\ 0 & 0 & 0 & 0 & -1/10\sqrt{5} & 0 & -1/10\sqrt{5} & 0 & 0 & 1/10\sqrt{5}+1/2 & 1/10\sqrt{5}+3/2 & 0 & 0 & 0 & 1/5\sqrt{5} & -1/5\sqrt{5} & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & -1/10\sqrt{5}-3/2 & 0 & -1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5} & 1/10\sqrt{5}+3/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/20\sqrt{5}+1/4 & 0 & -1/20\sqrt{5}-1/4 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 1/10\sqrt{5}+1/2 & 0 & -1/10\sqrt{5}-1/2 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/20\sqrt{5}+3/4 & 0 & -1/20\sqrt{5}-3/4 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & 0 & -1/10\sqrt{5}-3/2 & 0 & 1/10\sqrt{5}+3/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 0 & 1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5} & 0 & -1/10\sqrt{5}-3/2 & 0 & -1/10\sqrt{5}-1/2 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$1/5\sqrt{5}g_{29}+\left(-1/10\sqrt{5}+1/2\right)g_{24}+\left(1/10\sqrt{5}-3/2\right)g_{23}+1/5\sqrt{5}g_{18}+1/5\sqrt{5}g_{17}+1/5\sqrt{5}g_{16}+\left(-1/10\sqrt{5}+1/2\right)g_{12}+\left(1/10\sqrt{5}-3/2\right)g_{11}+\left(-1/10\sqrt{5}+1/2\right)g_{10}+\left(1/10\sqrt{5}-3/2\right)g_{9}+1/5\sqrt{5}g_{4}+1/5\sqrt{5}g_{2}+2h_{5}+\left(2/5\sqrt{5}-2\right)h_{3}+2h_{1}+1/5\sqrt{5}g_{-2}+1/5\sqrt{5}g_{-4}+\left(-1/10\sqrt{5}+1/2\right)g_{-9}+\left(1/10\sqrt{5}-3/2\right)g_{-10}+\left(-1/10\sqrt{5}+1/2\right)g_{-11}+\left(1/10\sqrt{5}-3/2\right)g_{-12}+1/5\sqrt{5}g_{-16}-1/5\sqrt{5}g_{-17}+1/5\sqrt{5}g_{-18}+\left(1/10\sqrt{5}-1/2\right)g_{-23}+\left(-1/10\sqrt{5}+3/2\right)g_{-24}-1/5\sqrt{5}g_{-29}$
matching e: $g_{29}+\left(-1/2\sqrt{5}+1/2\right)g_{23}+g_{16}+\left(1/2\sqrt{5}-1/2\right)g_{12}+\left(-1/2\sqrt{5}+1/2\right)g_{11}+\left(-1/2\sqrt{5}+1/2\right)g_{9}+\left(\sqrt{5}-1\right)h_{5}+\left(-\sqrt{5}+1\right)h_{3}+g_{-2}+\left(-1/2\sqrt{5}+1/2\right)g_{-10}+\left(1/2\sqrt{5}-1/2\right)g_{-11}+\left(-1/2\sqrt{5}+1/2\right)g_{-12}-g_{-17}+\left(1/2\sqrt{5}-1/2\right)g_{-24}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 1/5\sqrt{5}-1 & 1/5\sqrt{5}-3 & 0 & 0 & 0 & -2/5\sqrt{5} & -2/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5}+1 & -1/5\sqrt{5}+3 & 2/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 2/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5}+1 & -1/5\sqrt{5}+3 & 1/5\sqrt{5}-1 & 1/5\sqrt{5}-3 & 0 & -2/5\sqrt{5} & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0 & -1/5\sqrt{5}+1 & -1/5\sqrt{5}+3 & 0 & 0 & -2/5\sqrt{5} & -2/5\sqrt{5} & 0 & 1/5\sqrt{5}-1 & 1/5\sqrt{5}-3 & 2/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 1/10\sqrt{5} & 0 & 0 & 0 & -1/5\sqrt{5} & -1/10\sqrt{5}+1/2 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 1/10\sqrt{5}-1/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 0\\ 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 1/10\sqrt{5} & 0 & 0 & -1/10\sqrt{5}+3/2 & -1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}-3/2 & 0 & 0 & 1/5\sqrt{5} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}+3/4 & 1/20\sqrt{5}-3/4 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5}+2 & 0 & 0 & 1/10\sqrt{5}-3/2 & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}-1/2 & 0 & -1/5\sqrt{5}\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}+1/4 & 1/20\sqrt{5}-1/4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5}-2 & -1/10\sqrt{5}+1/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}+3/2 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/20\sqrt{5}+3/4 & 1/20\sqrt{5}-3/4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5}+3/2 & 1/5\sqrt{5}-2 & 0 & 0 & -1/5\sqrt{5} & 1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}+1/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/20\sqrt{5}+1/4 & 1/20\sqrt{5}-1/4 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 1/10\sqrt{5}-1/2 & 0 & 0 & -1/5\sqrt{5}+2 & 0 & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}-3/2 & 1/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/10\sqrt{5} & -1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & 0 & 1/10\sqrt{5}-3/2 & 0 & -1/5\sqrt{5} & 0 & -1/10\sqrt{5}+3/2\\ 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 0 & 1/10\sqrt{5} & 0 & 0 & -1/10\sqrt{5}+3/2 & -1/10\sqrt{5}+1/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 1/5\sqrt{5} & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5} & -1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 1/10\sqrt{5}-1/2 & 0 & 1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & -1/10\sqrt{5}+1/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}+3/4 & 0 & 1/20\sqrt{5}-3/4 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}+3/2 & 0 & 1/10\sqrt{5}-3/2 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}+1/4 & 0 & 1/20\sqrt{5}-1/4 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0 & 1/10\sqrt{5}-1/2 & 0 & -1/10\sqrt{5}+1/2 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/10\sqrt{5} & 0 & -1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & 0 & 1/10\sqrt{5}-1/2 & 0 & 1/10\sqrt{5}-3/2 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$1/5\sqrt{5}g_{29}+\left(-1/10\sqrt{5}-1/2\right)g_{24}+\left(1/10\sqrt{5}+3/2\right)g_{23}+1/5\sqrt{5}g_{18}+1/5\sqrt{5}g_{17}+1/5\sqrt{5}g_{16}+\left(-1/10\sqrt{5}-1/2\right)g_{12}+\left(1/10\sqrt{5}+3/2\right)g_{11}+\left(-1/10\sqrt{5}-1/2\right)g_{10}+\left(1/10\sqrt{5}+3/2\right)g_{9}+1/5\sqrt{5}g_{4}+1/5\sqrt{5}g_{2}-2h_{5}+\left(2/5\sqrt{5}+2\right)h_{3}-2h_{1}+1/5\sqrt{5}g_{-2}+1/5\sqrt{5}g_{-4}+\left(-1/10\sqrt{5}-1/2\right)g_{-9}+\left(1/10\sqrt{5}+3/2\right)g_{-10}+\left(-1/10\sqrt{5}-1/2\right)g_{-11}+\left(1/10\sqrt{5}+3/2\right)g_{-12}+1/5\sqrt{5}g_{-16}-1/5\sqrt{5}g_{-17}+1/5\sqrt{5}g_{-18}+\left(1/10\sqrt{5}+1/2\right)g_{-23}+\left(-1/10\sqrt{5}-3/2\right)g_{-24}-1/5\sqrt{5}g_{-29}$
matching e: $g_{23}+\left(1/2\sqrt{5}-1/2\right)g_{17}+g_{11}-g_{10}+g_{9}+\left(1/2\sqrt{5}-1/2\right)g_{4}+2h_{3}-2h_{1}-g_{-9}+g_{-10}+g_{-12}+\left(1/2\sqrt{5}-1/2\right)g_{-18}-g_{-24}+\left(-1/2\sqrt{5}+1/2\right)g_{-29}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 1/5\sqrt{5}+1 & 1/5\sqrt{5}+3 & 0 & 0 & 0 & -2/5\sqrt{5} & -2/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5}-1 & -1/5\sqrt{5}-3 & 2/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 2/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5}-1 & -1/5\sqrt{5}-3 & 1/5\sqrt{5}+1 & 1/5\sqrt{5}+3 & 0 & -2/5\sqrt{5} & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0 & -1/5\sqrt{5}-1 & -1/5\sqrt{5}-3 & 0 & 0 & -2/5\sqrt{5} & -2/5\sqrt{5} & 0 & 1/5\sqrt{5}+1 & 1/5\sqrt{5}+3 & 2/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 1/10\sqrt{5} & 0 & 0 & 0 & -1/5\sqrt{5} & -1/10\sqrt{5}-1/2 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 1/10\sqrt{5}+1/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 0\\ 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 1/10\sqrt{5} & 0 & 0 & -1/10\sqrt{5}-3/2 & -1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}+3/2 & 0 & 0 & 1/5\sqrt{5} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}-3/4 & 1/20\sqrt{5}+3/4 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & -1/5\sqrt{5}-2 & 0 & 0 & 1/10\sqrt{5}+3/2 & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}+1/2 & 0 & -1/5\sqrt{5}\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}-1/4 & 1/20\sqrt{5}+1/4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5}+2 & -1/10\sqrt{5}-1/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}-3/2 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/20\sqrt{5}-3/4 & 1/20\sqrt{5}+3/4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5}-3/2 & 1/5\sqrt{5}+2 & 0 & 0 & -1/5\sqrt{5} & 1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}-1/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/20\sqrt{5}-1/4 & 1/20\sqrt{5}+1/4 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 1/10\sqrt{5}+1/2 & 0 & 0 & -1/5\sqrt{5}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 1/10\sqrt{5}+3/2 & 1/5\sqrt{5}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/10\sqrt{5} & -1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & 0 & 1/10\sqrt{5}+3/2 & 0 & -1/5\sqrt{5} & 0 & -1/10\sqrt{5}-3/2\\ 0 & 0 & 0 & 0 & 1/10\sqrt{5} & 0 & 1/10\sqrt{5} & 0 & 0 & -1/10\sqrt{5}-3/2 & -1/10\sqrt{5}-1/2 & 0 & 0 & 0 & -1/5\sqrt{5} & 1/5\sqrt{5} & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/10\sqrt{5} & -1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 1/10\sqrt{5}+1/2 & 0 & 1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & -1/10\sqrt{5}-1/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}-3/4 & 0 & 1/20\sqrt{5}+3/4 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & -1/10\sqrt{5}-3/2 & 0 & 1/10\sqrt{5}+3/2 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/20\sqrt{5}-1/4 & 0 & 1/20\sqrt{5}+1/4 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 0 & 0 & 1/10\sqrt{5}+1/2 & 0 & -1/10\sqrt{5}-1/2 & 0 & 0 & 0 & 1/5\sqrt{5} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/10\sqrt{5} & 0 & -1/10\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & -1/5\sqrt{5} & 0 & 0 & 1/5\sqrt{5} & 0 & 1/10\sqrt{5}+1/2 & 0 & 1/10\sqrt{5}+3/2 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 0, 1), (-1, 0, -1), (0, 1, 1), (0, -1, -1), (1, 1, 1), (-1, -1, -1), (1, 0, 0), (-1, 0, 0), (0, 0, 1), (0, 0, -1), (0, -1, 0), (0, 1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144 & 0 & -72\\ 0 & 144 & -72\\ -72 & -72 & 144\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{5}_1$
h-characteristic: (0, 0, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 10
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $2A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 2: $5A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{2}_1+3A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $15V_{2\omega_{1}}+30V_{\omega_{1}}+31V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&5h_{8}+10h_{7}+10h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&g_{63}-g_{55}+g_{44}\\ f&=&g_{-44}-g_{-55}+g_{-63}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&5h_{8}+10h_{7}+10h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{63}-2g_{55}+2g_{44}\\ [h, f]&=&-2g_{-44}+2g_{-55}-2g_{-63}\end{array}$
Centralizer type: $C_3+B^{4}_2$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 31): $g_{-34}$, $g_{-28}$, $g_{-22}$, $g_{-21}$, $g_{-15}$, $g_{-14}$, $g_{-8}$, $g_{-7}$, $g_{-6}$, $h_{1}$, $h_{3}$, $h_{6}$, $h_{7}$, $h_{8}$, $g_{2}+g_{-16}$, $g_{4}+g_{-11}$, $g_{6}$, $g_{7}$, $g_{8}$, $g_{9}+g_{-10}$, $g_{10}+g_{-9}$, $g_{11}+g_{-4}$, $g_{14}$, $g_{15}$, $g_{16}+g_{-2}$, $g_{17}-g_{-23}$, $g_{21}$, $g_{22}$, $g_{23}-g_{-17}$, $g_{28}$, $g_{34}$
Basis of centralizer intersected with cartan (dimension: 5): $-h_{7}$, $-h_{6}$, $-h_{3}$, $-h_{1}$, $-h_{8}$
Cartan of centralizer (dimension: 5): $-h_{8}$, $-h_{7}$, $-h_{3}$, $-h_{1}$, $-h_{6}$
Cartan-generating semisimple element: $-4h_{8}-h_{7}-9h_{6}+7h_{3}+5h_{1}$
adjoint action: $\begin{pmatrix}9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -15/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 17/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -17/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 15/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5/2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{31}-887/2x^{29}+1357335/16x^{27}-146213985/16x^{25}+155333906175/256x^{23}-13066786283175/512x^{21}+2753009049105705/4096x^{19}-5406279602793135/512x^{17}+184738434183569835/2048x^{15}-376011054791899405/1024x^{13}+2449470653336134969/4096x^{11}-427722371192965041/1024x^9+943758161537625/8x^7-11381514322500x^5$
Factorization of characteristic polynomial of ad H: 1/4096(x )(x )(x )(x )(x )(x -9)(x -8)(x -8)(x -7)(x -6)(x -1)(x -1)(x +1)(x +1)(x +6)(x +7)(x +8)(x +8)(x +9)(2x -17)(2x -15)(2x -7)(2x -5)(2x -1)(2x -1)(2x +1)(2x +1)(2x +5)(2x +7)(2x +15)(2x +17)
Eigenvalues of ad H: $0$, $9$, $8$, $7$, $6$, $1$, $-1$, $-6$, $-7$, $-8$, $-9$, $17/2$, $15/2$, $7/2$, $5/2$, $1/2$, $-1/2$, $-5/2$, $-7/2$, $-15/2$, $-17/2$
31 eigenvectors of ad H: $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: C^{1}_3+B^{4}_2
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/9 & 0 & -1/18\\ 0 & 1/18 & -1/36\\ -1/18 & -1/36 & 1/18\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (3 total):
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{8}-4h_{7}-2h_{6}$
matching e: $g_{-28}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
$2h_{8}+2h_{7}$
matching e: $g_{15}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{8}-4h_{7}-2h_{6}$
matching e: $g_{-28}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
$2h_{8}+2h_{7}$
matching e: $g_{15}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (2, 1, 1), (-2, -1, -1), (1, 0, 0), (-1, 0, 0), (0, 1, 1), (0, -1, -1), (1, 0, 1), (-1, 0, -1), (2, 1, 2), (-2, -1, -2), (2, 0, 1), (-2, 0, -1), (0, 0, 1), (0, 0, -1), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}36 & 0 & -36\\ 0 & 72 & -36\\ -36 & -36 & 72\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36 & -1/72\\ -1/72 & 1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$2h_{3}-2h_{1}$
matching e: $g_{10}+g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$4h_{1}$
matching e: $g_{23}-g_{-17}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{3}-2h_{1}$
matching e: $g_{10}+g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$4h_{1}$
matching e: $g_{23}-g_{-17}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 0), (-1, 0), (1, 1), (-1, -1), (2, 1), (-2, -1), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144 & -144\\ -144 & 288\\ \end{pmatrix}$
Unfold the hidden panel for more information.

---

$A^{4}_1$
h-characteristic: (0, 0, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 8
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $2A^{2}_1$ Containing regular semisimple subalgebra number 3: $4A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $10V_{2\omega_{1}}+32V_{\omega_{1}}+42V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&4h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&g_{63}-g_{58}+g_{52}\\ f&=&g_{-52}-g_{-58}+g_{-63}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&4h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{63}-2g_{58}+2g_{52}\\ [h, f]&=&-2g_{-52}+2g_{-58}-2g_{-63}\end{array}$
Centralizer type: $C_4+2A^{4}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 42): $g_{-44}$, $g_{-39}$, $g_{-34}$, $g_{-33}$, $g_{-28}$, $g_{-27}$, $g_{-22}$, $g_{-21}$, $g_{-20}$, $g_{-15}$, $g_{-14}$, $g_{-13}$, $g_{-8}$, $g_{-7}$, $g_{-6}$, $g_{-5}$, $h_{1}$, $h_{5}$, $h_{6}$, $h_{7}$, $h_{8}$, $g_{2}+g_{-9}$, $g_{3}+g_{-3}$, $g_{5}$, $g_{6}$, $g_{7}$, $g_{8}$, $g_{9}+g_{-2}$, $g_{10}-g_{-16}$, $g_{13}$, $g_{14}$, $g_{15}$, $g_{16}-g_{-10}$, $g_{20}$, $g_{21}$, $g_{22}$, $g_{27}$, $g_{28}$, $g_{33}$, $g_{34}$, $g_{39}$, $g_{44}$
Basis of centralizer intersected with cartan (dimension: 5): $-h_{1}$, $-h_{8}$, $-h_{7}$, $-h_{6}$, $-h_{5}$
Cartan of centralizer (dimension: 6): $-h_{1}$, $-h_{7}$, $-g_{3}-g_{-3}$, $-h_{5}$, $-h_{8}$, $-h_{6}$
Cartan-generating semisimple element: $-9h_{8}+7h_{7}+5h_{6}-4h_{5}-h_{1}$
adjoint action: $\begin{pmatrix}4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -5/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -11/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 29/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 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0 & 0 & 0 & 0 & 0 & -13/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 27/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -25 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 17 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -23/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 21/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -29/2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 11/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{42}-3631/2x^{40}+21607429/16x^{38}-8850801979/16x^{36}+17941417849897/128x^{34}-5956190295349513/256x^{32}+5335665321390933873/2048x^{30}-12761043946089203955/64x^{28}+684969106288103718816525/65536x^{26}-48894690480221662828299335/131072x^{24}+9391937399782214200223510489/1048576x^{22}-149103906715668206437782123937/1048576x^{20}+764245467730495598587393204343/524288x^{18}-4878182178055599388748531774653/524288x^{16}+36474243757441439143589075071621/1048576x^{14}-71592505379114639070784054650237/1048576x^{12}+15114025086539364096060376545021/262144x^{10}-295347615249033957783930439725/16384x^8+471771104933468042843765625/256x^6$
Factorization of characteristic polynomial of ad H: 1/1048576(x )(x )(x )(x )(x )(x )(x -25)(x -17)(x -9)(x -8)(x -4)(x -3)(x -2)(x -1)(x +1)(x +2)(x +3)(x +4)(x +8)(x +9)(x +17)(x +25)(2x -29)(2x -27)(2x -23)(2x -21)(2x -13)(2x -11)(2x -7)(2x -5)(2x -1)(2x -1)(2x +1)(2x +1)(2x +5)(2x +7)(2x +11)(2x +13)(2x +21)(2x +23)(2x +27)(2x +29)
Eigenvalues of ad H: $0$, $25$, $17$, $9$, $8$, $4$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-4$, $-8$, $-9$, $-17$, $-25$, $29/2$, $27/2$, $23/2$, $21/2$, $13/2$, $11/2$, $7/2$, $5/2$, $1/2$, $-1/2$, $-5/2$, $-7/2$, $-11/2$, $-13/2$, $-21/2$, $-23/2$, $-27/2$, $-29/2$
42 eigenvectors of ad H: $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$
Centralizer type: C^{1}_4+2A^{4}_1
Reductive components (3 total):
Scalar product computed: $\begin{pmatrix}1/18 & -1/36 & -1/18 & 0\\ -1/36 & 1/18 & 0 & -1/36\\ -1/18 & 0 & 1/9 & 0\\ 0 & -1/36 & 0 & 1/18\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (4 total):
$-2h_{8}-4h_{7}-2h_{6}-2h_{5}$
matching e: $g_{-33}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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$2h_{8}+4h_{7}+4h_{6}+2h_{5}$
matching e: $g_{39}$
verification: $[h,e]-2e=0$
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$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
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$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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Linear space basis of intersection of centralizer and ambient Cartan:
$-2h_{8}-4h_{7}-2h_{6}-2h_{5}$
matching e: $g_{-33}$
verification: $[h,e]-2e=0$
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$2h_{8}+4h_{7}+4h_{6}+2h_{5}$
matching e: $g_{39}$
verification: $[h,e]-2e=0$
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$h_{8}+2h_{7}$
matching e: $g_{22}$
verification: $[h,e]-2e=0$
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$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1, 1), (-1, -1, -1, -1), (2, 2, 2, 1), (-2, -2, -2, -1), (1, 1, 1, 0), (-1, -1, -1, 0), (0, 0, 0, 1), (0, 0, 0, -1), (1, 0, 1, 0), (-1, 0, -1, 0), (1, 0, 2, 0), (-1, 0, -2, 0), (0, 0, 1, 0), (0, 0, -1, 0), (1, 0, 0, 0), (-1, 0, 0, 0), (2, 1, 2, 1), (-2, -1, -2, -1), (1, 1, 2, 1), (-1, -1, -2, -1), (1, 1, 0, 1), (-1, -1, 0, -1), (0, 1, 0, 1), (0, -1, 0, -1), (2, 1, 2, 0), (-2, -1, -2, 0), (1, 1, 2, 0), (-1, -1, -2, 0), (1, 1, 0, 0), (-1, -1, 0, 0), (0, 1, 0, 0), (0, -1, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}72 & -36 & -36 & 0\\ -36 & 72 & 0 & -36\\ -36 & 0 & 36 & 0\\ 0 & -36 & 0 & 72\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$g_{3}-2h_{1}+g_{-3}$
matching e: $g_{10}-g_{2}-g_{-9}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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Linear space basis of intersection of centralizer and ambient Cartan:
$g_{3}-2h_{1}+g_{-3}$
matching e: $g_{10}-g_{2}-g_{-9}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/36\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-g_{3}-2h_{1}-g_{-3}$
matching e: $g_{10}+g_{2}+g_{-9}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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Linear space basis of intersection of centralizer and ambient Cartan:
$-g_{3}-2h_{1}-g_{-3}$
matching e: $g_{10}+g_{2}+g_{-9}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}144\\ \end{pmatrix}$
Unfold the hidden panel for more information.

---

$A^{3}_1$
h-characteristic: (0, 0, 1, 0, 0, 0, 0, 0)
Length of the weight dual to h: 6
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 2: $3A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $6V_{2\omega_{1}}+30V_{\omega_{1}}+58V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&3h_{8}+6h_{7}+6h_{6}+6h_{5}+6h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&g_{63}-g_{58}\\ f&=&-g_{-58}+g_{-63}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&3h_{8}+6h_{7}+6h_{6}+6h_{5}+6h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{63}-2g_{58}\\ [h, f]&=&2g_{-58}-2g_{-63}\end{array}$
Centralizer type: $C_5+A^{8}_1$
Killing form square of Cartan element dual to ambient long root: 36
Basis of the centralizer (dimension: 58): $g_{-52}$, $g_{-48}$, $g_{-44}$, $g_{-43}$, $g_{-39}$, $g_{-38}$, $g_{-34}$, $g_{-33}$, $g_{-32}$, $g_{-28}$, $g_{-27}$, $g_{-26}$, $g_{-22}$, $g_{-21}$, $g_{-20}$, $g_{-19}$, $g_{-15}$, $g_{-14}$, $g_{-13}$, $g_{-12}$, $g_{-8}$, $g_{-7}$, $g_{-6}$, $g_{-5}$, $g_{-4}$, $h_{1}$, $h_{4}$, $h_{5}$, $h_{6}$, $h_{7}$, $h_{8}$, $g_{2}+g_{-9}$, $g_{4}$, $g_{5}$, $g_{6}$, $g_{7}$, $g_{8}$, $g_{9}+g_{-2}$, $g_{12}$, $g_{13}$, $g_{14}$, $g_{15}$, $g_{19}$, $g_{20}$, $g_{21}$, $g_{22}$, $g_{26}$, $g_{27}$, $g_{28}$, $g_{32}$, $g_{33}$, $g_{34}$, $g_{38}$, $g_{39}$, $g_{43}$, $g_{44}$, $g_{48}$, $g_{52}$
Basis of centralizer intersected with cartan (dimension: 6): $-h_{1}$, $-h_{8}$, $-h_{7}$, $-h_{6}$, $-h_{5}$, $-h_{4}$
Cartan of centralizer (dimension: 6): $-h_{5}$, $-h_{4}$, $-h_{1}$, $-h_{6}$, $-h_{8}$, $-h_{7}$
Cartan-generating semisimple element: $-11h_{8}+9h_{7}+5h_{6}-6h_{5}-7h_{4}-h_{1}$
adjoint action: $\begin{pmatrix}7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0 & 0 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Characteristic polynomial ad H: $x^{58}-13777/4x^{56}+20620443/4x^{54}-71988092033/16x^{52}+330282621520111/128x^{50}-530501084850796215/512x^{48}+309872266014324827137/1024x^{46}-269733569556969350049337/4096x^{44}+710883203293892364165562473/65536x^{42}-358134293727495491043782047753/262144x^{40}+17351144761434390556953892431401/131072x^{38}-2595370376980052114725557441895053/262144x^{36}+37489182331129556753904339898228523/65536x^{34}-3343221748517517378773885313621120409/131072x^{32}+57353361083140855201556245231090206243/65536x^{30}-3013736487308128707232842247571068316345/131072x^{28}+30105362063169372027599918110226244426457/65536x^{26}-1812005727135840019160995311625646057784597/262144x^{24}+10136821351477075333252933361198897327801049/131072x^{22}-165821824102369187103284920859257240276695297/262144x^{20}+242370050500481009647610316127182356146452831/65536x^{18}-245787034845069383446871080761228936660570375/16384x^{16}+166095925375019037893428988108050440090148125/4096x^{14}-17653797285834612798555116828516721435234375/256x^{12}+1077295498028247130562303028079518540234375/16x^{10}-31708282744240010604990792051193212890625x^8+4651770608904799666481611748062500000000x^6$
Factorization of characteristic polynomial of ad H: 1/262144(x )(x )(x )(x )(x )(x )(x -31)(x -21)(x -19)(x -16)(x -15)(x -12)(x -11)(x -10)(x -9)(x -7)(x -6)(x -5)(x -4)(x -4)(x -3)(x -2)(x -1)(x +1)(x +2)(x +3)(x +4)(x +4)(x +5)(x +6)(x +7)(x +9)(x +10)(x +11)(x +12)(x +15)(x +16)(x +19)(x +21)(x +31)(2x -35)(2x -27)(2x -15)(2x -11)(2x -7)(2x -5)(2x -3)(2x -3)(2x -1)(2x +1)(2x +3)(2x +3)(2x +5)(2x +7)(2x +11)(2x +15)(2x +27)(2x +35)
Eigenvalues of ad H: $0$, $31$, $21$, $19$, $16$, $15$, $12$, $11$, $10$, $9$, $7$, $6$, $5$, $4$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-4$, $-5$, $-6$, $-7$, $-9$, $-10$, $-11$, $-12$, $-15$, $-16$, $-19$, $-21$, $-31$, $35/2$, $27/2$, $15/2$, $11/2$, $7/2$, $5/2$, $3/2$, $1/2$, $-1/2$, $-3/2$, $-5/2$, $-7/2$, $-11/2$, $-15/2$, $-27/2$, $-35/2$
58 eigenvectors of ad H: $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: C^{1}_5+A^{8}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/18 & -1/36 & 0 & 0 & -1/36\\ -1/36 & 1/18 & 0 & -1/36 & 0\\ 0 & 0 & 1/9 & -1/18 & 0\\ 0 & -1/36 & -1/18 & 1/18 & 0\\ -1/36 & 0 & 0 & 0 & 1/18\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (5 total):
$2h_{8}+4h_{7}+4h_{6}+2h_{5}+2h_{4}$
matching e: $g_{43}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 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$-2h_{8}-4h_{7}-2h_{6}-2h_{5}-2h_{4}$
matching e: $g_{-38}$
verification: $[h,e]-2e=0$
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$h_{8}+2h_{7}+2h_{6}+2h_{5}$
matching e: $g_{44}$
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$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
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Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{8}+4h_{7}+4h_{6}+2h_{5}+2h_{4}$
matching e: $g_{43}$
verification: $[h,e]-2e=0$
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$-2h_{8}-4h_{7}-2h_{6}-2h_{5}-2h_{4}$
matching e: $g_{-38}$
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$h_{8}+2h_{7}+2h_{6}+2h_{5}$
matching e: $g_{44}$
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$-2h_{8}-2h_{7}-2h_{6}$
matching e: $g_{-21}$
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Elements in Cartan dual to root system: (1, 1, 1, 1, 1), (-1, -1, -1, -1, -1), (2, 2, 2, 2, 1), (-2, -2, -2, -2, -1), (1, 2, 2, 2, 1), (-1, -2, -2, -2, -1), (1, 1, 2, 1, 1), (-1, -1, -2, -1, -1), (1, 1, 0, 1, 1), (-1, -1, 0, -1, -1), (1, 0, 0, 0, 1), (-1, 0, 0, 0, -1), (1, 1, 1, 1, 0), (-1, -1, -1, -1, 0), (0, 0, 0, 0, 1), (0, 0, 0, 0, -1), (1, 2, 2, 2, 0), (-1, -2, -2, -2, 0), (0, 1, 1, 1, 0), (0, -1, -1, -1, 0), (1, 1, 2, 1, 0), (-1, -1, -2, -1, 0), (1, 1, 0, 1, 0), (-1, -1, 0, -1, 0), (0, 1, 2, 1, 0), (0, -1, -2, -1, 0), (0, 1, 0, 1, 0), (0, -1, 0, -1, 0), (1, 0, 0, 0, 0), (-1, 0, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, -1, 0, 0), (0, 0, 1, 1, 0), (0, 0, -1, -1, 0), (1, 1, 2, 2, 1), (-1, -1, -2, -2, -1), (1, 1, 0, 0, 1), (-1, -1, 0, 0, -1), (1, 1, 2, 2, 0), (-1, -1, -2, -2, 0), (0, 1, 2, 2, 0), (0, -1, -2, -2, 0), (1, 1, 0, 0, 0), (-1, -1, 0, 0, 0), (0, 0, 2, 1, 0), (0, 0, -2, -1, 0), (0, 0, 0, 1, 0), (0, 0, 0, -1, 0), (0, 1, 0, 0, 0), (0, -1, 0, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}72 & -36 & 0 & 0 & -36\\ -36 & 72 & 0 & -36 & 0\\ 0 & 0 & 36 & -36 & 0\\ 0 & -36 & -36 & 72 & 0\\ -36 & 0 & 0 & 0 & 72\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/72\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-4h_{1}$
matching e: $g_{2}+g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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Linear space basis of intersection of centralizer and ambient Cartan:
$-4h_{1}$
matching e: $g_{2}+g_{-9}$
verification: $[h,e]-2e=0$
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Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}288\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{2}_1$
h-characteristic: (0, 1, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 4
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $2A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{2\omega_{1}}+24V_{\omega_{1}}+79V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&2h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}\\ e&=&g_{64}+g_{62}\\ f&=&g_{-62}+g_{-64}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&2h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{64}+2g_{62}\\ [h, f]&=&-2g_{-62}-2g_{-64}\end{array}$
Centralizer type: $C_6$
Unfold the hidden panel for more information.

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$A_1$
h-characteristic: (1, 0, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 2
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{2\omega_{1}}+14V_{\omega_{1}}+105V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}\\ e&=&g_{64}\\ f&=&g_{-64}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}\\ [h, e]&=&2g_{64}\\ [h, f]&=&-2g_{-64}\end{array}$
Centralizer type: $C_7$
Unfold the hidden panel for more information.

---